GNOME is part of the GNU project, so the only "pressure" they could apply to GNOME is to make their enterprise offering entirely free software (at which point they will run it themselves because of GNUs and the FSFs position on Service-as-a-Software-Substitute[1]).

[1]: https://www.gnu.org/philosophy/who-does-that-server-really-s...

I talked what would happen if GNOME was hosted on the (currently proprietary) GHE as OP proposed.

Ah okay, I misunderstood. ;)