Nonfree local modules detect algebra automorphisms

Let $A$ be an algebra in a monoidal category, and let $M$ be a local $A$-module. Say that $M$ is free if it is a free right $A$-module.

Automorphism-detection conjecture. If $M$ is not free, then $M$ cannot be fixed by all algebra automorphisms of $A$.

The claim complements the preceding observation that free right $A$-modules are fixed by every algebra automorphism. No proof or resolution is supplied in the source.