A note on projective dimension over twisted commutative algebras

Let $M$ be a finitely generated module over a free twisted commutative algebra $A$ that is finitely generated in degree one. We show that the projective dimension of $M({\mathbf{C}}^n)$ as an $A({\mathbf{C}}^n)$-module is eventually linear as a function of $n$. This confirms a conjecture of Le, Nagel, Nguyen, and Römer for a special class of modules.