Well I get that, but I'm curious what would be the "government employees' work product" in this case, using the Supreme Court's chosen phrase. The case you linked is about someone doing their job in a way that his bosses thought was wrong. It just happened to involved speech. I don't see the connection to this case. To the extent that the criticism of the politician was related to that person's work duties, it's because they're paid by a public university to share opinions related to their expertise.

The work product of a state university professor is, in part, their lectures.

I think https://en.wikipedia.org/wiki/Demers_v._Austin has it right in this regard - that public educators have a specific First Amendment need/exception to be able to teach facts their chain of command doesn't appreciate - but it doesn't yet apply in Texas.