This isn't going anywhere unless they get support form outside of that group. The group has 25 members. Braverman is a member of the ERG with 43 members. One of which (Jonathan Gullis) is a member of both. So that is a potential 77 votes if she garners full support.
There is also the NZSG where I cannot find a list of members. I am not surprised really. Having to explain to constituents that you want to burn the planet doesn't bring the votes these days.