Senate Majority Leader John Thune and House Speaker Mike Johnson during a news conference in Statuary Hall at the US Capitol on Friday. Graeme Sloan/Bloomberg/Getty Images

The shutdown messaging war is in full effect, and early indications are that Republicans have been put in something of a bind , given the Democrats’ demand to extend enhanced Obamacare subsidies appears quite popular.

So Republican leaders have returned to a very familiar strategy – making false generalizations and misleading claims about undocumented immigrants, in an attempt to center the debate on more favorable terrain.

They’ve argued over and over again that Democrats are trying to provide health care to undocumented immigrants.

In fact, as CNN’s Tami Luhby has fact-checked , the changes Democrats are seek

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