With the divided Congress stalled at loggerheads on a multitude of partisan issues, House Democrats have tried to rejuvenate support for their agenda with seemingly every procedural gambit in the books. But so far with every overture, Republicans have calmly resisted the tendency to cave to Democratic demands.

In the past several weeks, Democrats have attempted to force votes on the Paycheck Fairness Act, an extension of emergency unemployment benefits and an increase to the federal minimum wage. Wednesday, they will add immigration overhaul to the list of stalemates they hope to break through the arcane procedures of the House.

"This is a way to allow the 20-something Republicans who have publicly said that they support comprehensive reform to join with Democrats and get this on the floor of the House," Rep. Joaquin Castro, a freshman Democrat from Texas, explained.

Even if all 199 House Democrats sign onto the petition, at least 19 Republicans would have to buck their party - an unlikely feat - to force House Speaker John Boehner's hand. Senior Democratic aides concede that the ploy is little more than something for supporters to rally around during the legislative lull.

Rep. Ileana Ros-Lehtinen, a Florida Republican who has vocally expressed support for comprehensive overhaul, today told ABC News that she will not sign the Democrats' discharge petition, even though it would take conservatives like her to break the impasse.

"I don't sign discharge petitions any more than the Democrats signed Republican discharge petitions when they were in the majority," she said. "So I guess we could ask the Democrats how many they signed when they were in the majority, which is they don't."

Ros-Lehtinen admitted that discharge petitions are "a good tool to use to get attention and to advance the cause" but, she added, the majority party rarely signs them when they're in power.

Castro, who has played a vocal role as his party's attempts to push comprehensive immigration overhaul though Congress, says the maneuvers are not all just for show.

"Forget about whether it's a Democratic thing or a Republican thing," Castro said. "It's the right thing to do. It's what America wants us to do, and I hope that we can get their support to do it."

Without a vote by the end of the current term of Congress, the Senate's historic effort to enact an overhaul will expire and legislators would have to return to square one.

Still, Castro is optimistic Democrats can get a bill to President Obama's desk.

"The only thing it will take for Republicans to put that bill on the floor is for them to decide that they're going to listen to the American people and do what the American people are asking us as a Congress to do," he said. "I'm an eternal optimist."

Ros-Lehtinen is also hopeful Congress will reach an agreement before the next session is seated.

"I'm sure people say that about every discharge petition - 'that's the only way' - but I'm still optimistic that we can get immigration reform done," she said. "I'm not a pessimist. I'm in the hope business."