Asymptotic regularity bound for symbolic powers of reduced ideal sheaves

Let $\mathscr{I}$ be an ideal sheaf defining a reduced subscheme $Z$ of $\mathbb{P}^n$, and let $s=s(\mathscr{I})$ be the $s$-invariant.

Asymptotic regularity conjecture. There is a constant $e$ such that for all $p\geq 1$, one has

$$\operatorname{reg} \mathscr{I}^{(p)}\leq sp+e.$$

The preceding results establish this type of linear bound under additional hypotheses, including when the non-locally-complete-intersection locus has dimension at most zero. The conjecture proposes that the same bound holds for symbolic powers of every ideal sheaf defining a reduced subscheme of projective space.