Eventual linearity of projective dimension for FI- and OI-modules

Let $c$ be a positive integer. Let $\mathbf{M}$ be a finitely generated $\operatorname{OI}$-module over $\mathbf{P}=({\mathbf{X}}^{\operatorname{OI},1})^{\otimes c}$, or a finitely generated $\operatorname{FI}$-module over $\mathbf{P}=({\mathbf{X}}^{\operatorname{FI},1})^{\otimes c}$. Projective-dimension growth conjecture. There are integers $A,B$ such that

$$\operatorname{pd}_{\mathbf{P}_n}\mathbf{M}_n=An+B$$

whenever $n\gg0$. The paper presents this as an expectation about the number of columns in the Betti tables; the supplied text gives no resolution evidence, so its status remains open.