Latest "John-Wells" stories
The measure had been stalled on Capitol Hill for several weeks, as court moves took precedence for advocates.
The high court extended the appeals deadline on the issue for 30 days, although Department of Justice officials have not said if they will pursue one.
Both leaders of the House Veterans' Affairs Committee are trying to revive the legislative effort that stalled last month.
Four former VA leaders are urging lawmakers to reject the plan, while former VA Secretary David Shulkin is arguing for it.
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