On existence of finitely generated projective generator with commutative endomorphism ring in ${}_R Mod$

Let $$R$$ be a ring with unity (not necessarily commutative). Let $${}_R Mod$$ be the category of left $$R$$-modules. If $${}_R Mod$$ has a projective generator with commutative endomorphism ring , then does $${}_R Mod$$ have a finitely generated projective generator with commutative endomorphism ring ?

• I still don’t know the answer, but I raised exactly the same question in a comment on this MSE question some time ago: math.stackexchange.com/questions/1364337/…. It would be interesting to know whether your interest in the question is for similar reasons. Jan 1, 2019 at 10:17
• @JeremyRickard: I just saw your comment in the linked question ... and yes I was also thinking about how to distinguish the category of modules over a commutative ring from that over a general ring .. Jan 2, 2019 at 5:55