Updated 2:30 p.m. | As Heritage Action for America pushes lawmakers to kill any bill that funds Obamacare, the conservative group is running into a question it refuses to answer: If the government shuts down, what next?

In private meetings with Heritage Action representatives — namely, Heritage Action CEO Mike Needham, Chief Operating Officer Tim Chapman and Director of House Relations Erin Siefring — lawmakers say they have asked Heritage Action what its strategy would be if the government shut down.

And Heritage Action, they say, does not have an answer.

“There is no plan B, there is no ‘what if,’” said one Republican lawmaker who spoke to CQ Roll Call on the condition of anonymity. He said he had talked to a “handful” of members who met with Heritage Action and the outside conservative group had “no viable alternative” to the president caving in.

“That’s leaving us in the lurch. That’s not proper planning,” the lawmaker said, explaining that the president defunding his administration’s signature piece of legislation was not exactly a likely scenario.

On Wednesday, Speaker John A. Boehner laid out a strategy of “targeted strikes” against the health care law to highlight “significant Democratic defections, chipping away at the legislative coalition that keeps the president’s health care law on the books,” according to a GOP leadership aide. The scheme runs in direct contrast with the tactic of defunding the health care law in a continuing resolution that some conservatives are pushing for but others dismiss as a pipe dream.

According to a source in the room, Boehner advocated for a strategy of picking apart the health care law piece by piece.

“We should view the delay votes this month as the opening salvo in a series of well-placed, targeted strikes that will ultimately dissolve the Obamacare coalition and topple this train wreck of a law,” Boehner said, according to the source.

Calling it “our best shot at actually getting rid of Obamacare,” the Ohio Republican’s gambit falls well shy of the approach outside groups such as Heritage are advocating. They want Republicans to vow to vote against any CR that funds Obamacare.

Although Boehner offered a piecemeal strategy, he twice refused to rule out threatening a government shutdown over Obamacare in a news conference with reporters.

“No decisions have been made on the CR,” he said.

But the speaker has never been enamored of threats to shut down the government over Obamacare, and many of his allies have spoken out sharply in recent days against the idea.

Heritage Action’s communications director, Dan Holler, said Republicans were giving up on the idea of defunding Obamacare, lacking imagination and mired in a question impossible to answer.

“You’re asking a hypothetical because you don’t know what the playing field is going to look like,” Holler told CQ Roll Call late Tuesday night. “You don’t know what the playing field is going to look like at the end of September.”

The government is funded through Sept. 30. After that, a stopgap measure is needed to keep the lights on. Heritage has been lobbying members to vow to vote against any such bill that doesn’t defund Obama’s health care law — and they say the five-week August recess is a great time to start changing the outside dynamics.

Holler said GOP leaders are “afraid of pushing” members because they fear the feat is impossible. “We have no idea what’s possible,” Holler said.

But the Republican lawmaker said that’s precisely the problem: “It’s all the art of the possible.”

“This is a laudable goal,” the lawmaker said. “But how can we get this president to meet us halfway on something?”

In response to the lawmaker’s comments, Holler said when House conservatives “think about the ‘art of the possible’ they should be thinking about setting the agenda, driving a message and defining the playing field, not analyzing the chalk lines on Obama’s home turf.”

Still, the Republican lawmaker contended that Heritage had not “thought that deeply beyond” the first stage of its plan.

If a continuing resolution defunding Obamacare somehow — miraculously — got through the Senate, the president would veto it. The more likely scenario is that such a bill would never reach the president’s desk, and Democrats and the White House would bash Republicans daily for playing politics with a simple bill to keep the government running.

Aides on both sides of the aisle theorize that it would be Republicans shouldering most of the blame for a government shutdown, and Obama would have vast political incentive to wait for the GOP to surrender.

A senior Republican aide said if the government shuts down in an effort to defund Obamacare, “it’s not hypothetical to know we’ll need a plausible next step or risk harming the overall cause to rid this country of that terrible law.”

Another senior GOP aide contended, “If outside groups like Heritage Action are trying to sell this strategy, then publicly attacking good conservatives with reasonable questions as pro-Obamacare or not committed to its repeal seems disingenuous.”

The Republican lawmaker had perhaps the most damning indictment of Heritage Action; he said the defund Obamacare strategy “probably fits nicely with their fundraising metrics.”

But Holler insists this effort is about the GOP’s last chance to stop implementation of the health care law. He said this was leadership’s opportunity to show the nation what conservatives really stand for.

“We’d love to be pushing with them to defund Obamacare, and if they did, we’d be side by side with them,” Holler said. “And it’d be awesome.”