Four years after Obamacare was enacted, and more than 50 House votes to undo it, Republicans remain dedicated to destroying the law. But they’re still lost on what they’d put in its place if given the chance.

House Majority Leader Eric Cantor (R-VA) is leading the effort to craft a GOP alternative, and promised his members a vote in 2014. He faces a sea of obstacles to writing a health care bill with sufficient support in the House, and potentially a world of hurt if he follows through with his commitment.

“House Republicans will rally around and pass an alternative to Obamacare this year,” Cantor told Republicans at their annual conference retreat in January.

The aim is to stitch together a series of ideas with broad GOP support, such as letting people buy insurance across state lines, reforming medical malpractice laws and expanding high-risk pools and incentive-based health savings accounts, according to sources familiar with the effort.

Cantor is said to be holding regular meetings with House Majority Whip Kevin McCarthy (R-CA) and House GOP Conference Chair Cathy McMorris Rodgers (R-WA), and roping in the House doctors caucus and the ultraconservative Republican Study Committee. One of the reasons he believes it’s possible is that hard-core conservatives, the sort that most often scuttle leadership’s plans, actually want an alternative.

But it’s still a long shot with high risks and few rewards.

For one, the project is greatly complicated by election year considerations. Many Republicans would rather hammer Obamacare than do the hard work of crafting and defending a bill of their own. And even if House Republicans coalesce around a bill — a big if — it’ll invariably carry its own set of tradeoffs and disruptions. That’ll be tougher to defend while Republicans are raising hell about cancellations and other disruptions caused by Obamacare.

Simply repealing Obamacare would, at this juncture, take away coverage from upwards of 10 million Americans who are benefiting from the law’s insurance subsidies, Medicaid expansion or the provision that permits children to stay on their parents’ policies.

And their own ideas would have sweeping impacts, too. Health policy experts say allowing insurers to sell plans across state lines, for example, would upend the system by motivating insurers to cancel policies and relocate to states with the fewest consumer protections. Any such bill would be scored by the Congressional Budget Office, and that would force Republicans to answer for the downsides.

Replacing Obamacare isn’t a priority for Republicans, and will continue to take a back seat to their efforts to dismantle and chip away at the law. GOP statements in advance of the Sunday anniversary were heavy on attacks and scant on what to do instead. The party is eying probable gains in the 2014 mid-term elections and Obamacare’s persistent unpopularity hurts Democrats. Cantor’s memo to Republicans on Friday about the upcoming work period made no mention of an Obamacare alternative.