Mayor Kirk Caldwell and his mayoral challenger, Charles Djou, have traveled very different paths since 2005 to get to the same conclusion: Honolulu’s elevated rail line, both of them now say, must be built to Ala Moana Center.

But the candidates still disagree sharply over how to pay for the severely cash-strapped project.

Despite a stumble in his messaging this summer, Caldwell insists that he remains steadfast in his support of the full rail line even as its costs have swelled by at least $3.3 billion in the past two years. He’s said he’ll go before state lawmakers again next year and ask for another extension of the rail tax to help close a massive budget gap of at least $1.8 billion.

“What gets you far is the excise tax, some kind of sharing with the state,” Caldwell said earlier this month at his Iwilei campaign headquarters. “I’m already sitting down with the leaders … to talk about some form of extension that is fair to everyone, including the state Legislature. And I think there is opportunity there. I’m hearing it already.”

There’s no guarantee that Caldwell’s gambit at the state Capitol will work. Meanwhile, Djou’s own uncertain, alternative solution would rely on successes across multiple fronts.

Djou said he’s optimistic that reforming rail “mismanagement,” conducting a financial audit, pursuing the 10 percent fee the state collects on Oahu’s general excise tax surcharge, creating partnerships with private entities on the rail line, enacting fees on developers and asking rail’s federal partners to reconsider their decision to withhold more funding should be enough to close the nearly $2 billion gap.

“You add that all up, will we be able to get there? Well, I haven’t had my audit yet. But I do think there’s a hunk of change there that can get us to where we want to go,” the former Hawaii congressman and city councilman said Friday.

Even if all of those steps don’t succeed, Djou said Friday that he would not pursue — nor would he approve — a GET surcharge extension to finish rail.

In previous weeks it had been difficult to pin down his position on the issue. On Sept. 1 Djou had expressed a tepid willingness to consider asking for such a rail tax extension — but only “as a last resort” after taking the other steps. Later that month he told Hawaii News Now’s “Sunrise” show that he would not increase taxes for rail, “period.”

Caldwell, who’s also pursuing public-private partnerships and a share of the state’s administrative fees, which are often called the “skim,” has criticized Djou’s rail stance as unrealistic. “I have a plan. He doesn’t have a plan,” Caldwell said at his headquarters.

Djou, meanwhile, criticized Caldwell on Friday for pursuing a tax extension before taking the steps that Djou has proposed first — moves that he said would go a long way toward restoring public trust in the transit project.

Both men also said they oppose using property taxes to finish building rail.

In the past decade Caldwell has embraced building rail while Djou has staunchly opposed it.

In 2005, when Oahu’s fourth effort in three decades to build a mass-transit system started to take off, Caldwell served as a state representative, and Djou served on the City Council. That year, Caldwell helped to introduce and then voted for House Bill 1309, the first step that allowed the counties to levy a surcharge for new transportation projects. It later became law.

When the issue then came before the Honolulu City Council, Djou and Barbara Marshall were the only two members to vote against Bill 40, which the nine-person body enacted to create the 0.5 percent general excise surcharge that’s funded most of rail so far.

Then-Councilman Djou would vote against other rail measures. One of them was Bill 78 in 2006. It affirmed the city’s decision to select a fixed guideway from Kapolei to the University of Hawaii at Manoa as its locally preferred alternative for mass transit instead of managed lanes.

Djou did join the unanimous 2009 Council vote to create the Honolulu Authority for Rapid Transportation, the semi-autonomous agency overseeing rail’s completion.

“I said it was going to take way longer, cost way more and not resolve our traffic problems,” Djou told the Honolulu Star-Advertiser’s editorial board in July. “And here I am, a few years later … and everything that I said would happen has happened. But we’re stuck.”

This summer he campaigned on exploring major changes to the transit system to make it affordable on its existing $6.8 billion budget, such as converting it to a street-level light-rail system past Middle Street.

After the FTA made clear in late August that it would revoke more than $1.5 billion in rail funding if the city stopped building at Middle Street, Djou said he believed that left no choice but to go all the way to Ala Moana.

“We’re going to have to do rail,” Djou told the Star-Advertiser on Sept. 1, a day after meetings with the Federal Transit Administration on the project’s future wrapped up in San Francisco. “You cannot stop it at Middle Street.”

Meanwhile, Caldwell spent part of the summer reaffirming his support for the full 20-mile, 21-station rail line after having said the project should halt at Middle Street — for now.

On June 16, facing what was then an Aug. 7 deadline to submit a rail “recovery” plan to the FTA, Caldwell told the Honolulu Authority for Rapid Transportation board that the tight time frame left the city no flexibility to come up with a real strategy. Rail would have to “live within the bounds” of its existing but insufficient funding, he said.

“I have to say, I wish we could go all the way to Ala Moana now, and I really wish we could up to UH Manoa,” Caldwell told the HART board. “But that’s for another day, in my mind.

“I’m totally committed to going further. But let’s focus on what we can do with the money we have,” the mayor added. “It doesn’t preclude future extensions. Let’s do a good job for the first 15 miles, and then we can talk about the rest later.”

The comments helped spark more uncertainty over the project’s future. Caldwell said the comments were misconstrued in the media.

“Maybe I didn’t say it clearly,” he told the Star-Advertiser board a month later. “I could have said it more clearly. With the money we’ve got, we can only get to Middle Street. It doesn’t mean we’re stopping.”

After Caldwell made his initial comments in June, the FTA extended the deadline to Dec. 31 and is currently considering extending it further to next summer.

That would leave time after the election for rail leaders to plead their case before state lawmakers for a GET extension.