Eventual linearity of regularity for FI- and OI-modules

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

$$\operatorname{reg} \mathbf{M}_n=an+b$$

whenever $n\gg0$. This is known for monomial FI-ideals when $c=1$, while it remains open for monomial OI-ideals of $\mathbf{X}^{\operatorname{OI},1}$; more generally, only linear upper bounds and special cases are established.