  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 $M({\bf C}^n)$ as an $A({\bf C}^n)$-module is
eventually linear as a function of $n$. This confirms a conjecture of Le,
Nagel, Nguyen, and R\"omer for a special class of modules.
