( CNN ) -- The government 's open . Washington is back at work . House Republicans , licking their wounds , are asking themselves what 's next . And President Barack Obama has thrown down the gauntlet : The top item on his agenda is immigration reform . What are the chances that the House will now move ahead on immigration ? The answer will have less to do with immigration than with how the budget battle has changed the larger political dynamic in Washington . House Republicans ' views on immigration are untested , and many advocates for reform believe they are implacably hostile . But the truth is Republican opinion has been evolving since the 2012 election . More and more House Republicans , perhaps the majority , know that reform is overdue and that the GOP must be part of the solution -- to remain competitive with Latino voters and because it 's the right thing to do . Individual lawmakers and essential staff continued to work on the issue even through the dark days of the shutdown . And members are coalescing around answers to the hardest of the hard questions : what to do about immigrants living in the United States illegally ? House Majority Leader Eric Cantor is working on a bill that would create a path to citizenship for `` Dreamers '' brought to the U.S. illegally as children . And one recent informal count found 84 House Republicans -- more than one third of the total -- in favor of legal status for the Dreamers ' parents . Bottom line : If it were n't for the rancor of the budget brawl , the House might be in a good place on immigration , with Republicans ready to move forward and pass a package of measures they could send to a conference with the Senate bill . Opinion : Key to immigration reform -- worker visas . So what exactly is the fallout from the budget battle ? Surprisingly , it appears to cut both ways -- both for and against the prospect of an immigration overhaul . Even before the government reopened , two different factions were making their voices heard . Some , such as Rep. Raul Labrador of Idaho -- an opponent of the budget deal but a strong proponent of immigration reform -- argued that the budget battle had made it hard , if not impossible , for House Republicans to reach a deal with Obama . Others , such as Rep. Aaron Schock of Illinois , who voted for the budget package , said it 's time to get back to the give and take of governing -- time to sit down with Democrats and compromise , including on immigration . Opinion : It 's not Syria holding up immigration reform . Which of these two sentiments will prove stronger in the House ? GOP lawmakers are reeling from their recent drubbing . Skepticism and negativity are at an all-time high . And it will take both kinds of champions -- tough-minded and accommodating -- to negotiate a deal . But if enough other Republicans agree with Schock and make their views known , that could empower leadership to open the way to consideration of some immigration bills . A handful of hard-line conservatives -- the group that opposed the budget deal -- still hold enormous sway in the House . And just because Speaker John Boehner waived the so-called Hastert rule once -- bringing the legislation that ended the shutdown up for a vote when he knew it lacked support from the majority of the Republican majority -- does n't mean he 'll do that again anytime soon . The House Republican conference is only as strong as it is cohesive , and the majority-of-the-majority rule has proved a good way to maintain that power and cohesion . Still , the complex dynamic that drives Republicans in the House may have shifted somewhat in the shutdown . Certainly you hear a lot more grumbling , in private and in public , about the power of the hard-line naysayers . Other members are tired of being held hostage . Many want to get on with governing , making deals on a wide range of issues . And a few , such as Schock , are starting to say so , even on TV . Opinion : How Obama can clinch immigration deal . That 's a ray of hope . But there 's still another danger looming . The one thing House Republicans are not going to do in the wake of the budget battle -- not on any issue , in any circumstances -- is a favor for Obama . And to the degree that immigration reform is seen as Obama 's issue , it will be dead on arrival in the House . The question for House Republicans , leadership and rank and file : Do they want to cede the issue to Obama ? Can they afford to let him own it ? More and more of the GOP grasps that that 's a mistake . It 's a disaster politically for the party and a mistake for the nation , which needs reform , urgently , for the sake of the economy and the rule of law . What 's needed in the House now is not a favor for Obama , but a strong conservative answer on one of the most vexing issues facing the nation . Will House Republicans see it that way ? Can they take ownership and move forward ? The opinions expressed in this commentary are solely those of Tamar Jacoby .
Tamar Jacoby : After budget battle , GOP may be in no mood for immigration overhaul . But many in GOP favor legal status and have been working on reforms , she says . She says others refuse to do President Barack Obama any favor by addressing issue . Jacoby : GOP must take ownership , provide a strong conservative approach to reform .