Non-Binding Anonymous Conscience Vote

A story from This American Life implied that there are several Republican House members that are secretly supporters of efforts to tackle Climate Change.....  What about a Non-Binding Anonymous Conscience Vote then?  Would it help move the issue forward if everyone voted their conscience on issue without fear of reprisal or commitment?

Preliminary Rules:

1) The House Minority Leader or official equivalent may call for a Non-Binding Anonymous Conscience Vote on any issue once per legislative year.

2) No House member may be allowed to indicate or convey in any manner how they voted.