The opt-out movement is one of the biggest storylines these days in education policy. But so far, it hasn’t really been a key issue as Congress wrestles with a reauthorization of the Elementary and Secondary Education Act.

That could change, as of this week. The U.S. House of Representatives, which yanked its GOP-only rewrite of the law earlier this year in the wake of conservative opposition, is expected to bring the bill back up for a vote this week, just as the Senate considers its bipartisan, carefully balanced compromise legislation.

In order to get its bill through the chamber this time, House GOP leaders are likely to let lawmakers vote on the A-Plus Act, which would allow states to largely opt out of federal accountability as long as they come up with their own plans for gauging schools’ progress. It probably won’t pass, but conservative lawmakers will be happy to get themselves on the record as supporting the legislation and might be more willing to vote for the bill.

But there could be another amendment in the mix, introduced by Rep. Matt Salmon, R-Ariz. It would allow parents to opt their children out of state standardized tests, without penalizing the school for accountability purposes. Right now, under the law’s current version, the No Child Left Behind Act, 95 percent of students must participate in testing, or the school is considered as not meeting achievement targets.

Opt-out has been a huge headache for the Obama administration lately. Oregon recently passed legislation smoothing the way for parent opt-outs. And Colorado tried to wiggle out of the 95 percent requirement in its waiver renewal application, to no avail.

It’s easy to see how the amendment—if it’s allowed to be offered—would generate bipartisan support. At one point, it looked like Congress might limit the number of tests mandated under the NCLB law (that’s annual tests in reading and math in grades 3 through 8 and once in high school, plus science tests in certain grades). That idea got nixed in order to bring forward a bipartisan compromise, but the anti-testing crowd might see this amendment as the next best thing.

And it’s equally easy to see how this might make the civil rights community, which already hates the House bill and isn’t wild about the Senate’s bipartisan legislation, really uneasy. After all, couldn’t a school encourage parents of, say, English-language learners who might not perform so well on assessments, to keep them home on testing day?

There could be conservative fans of opt-out over on the Senate side, too. After all, two GOP presidential candidates, Sens. Ted Cruz of Texas and Marco Rubio of Florida, supported a bill that would prohibit the federal government from interfering with local decision-making on tests.

Other amendments that could come up for a vote in the House: an amendment to shorten the authorization of the bill (an idea that failed to gain traction earlier this year) and an amendment making crystal clear, once and for all, that the U.S. Department of Education cannot ever, ever under any circumstances force or encourage or even mildly suggest that a state adopt Common Core State Standards. That’s already in the bill, in the legislative equivalent of big bold letters, but voting on it as a standalone issue could make conservatives happier and bolster the bill’s chances.