The regularity bound for the stabilization point of the minimum distance function

Let $S=\mathbb{K}[x_1,\ldots,x_n]$ be a polynomial ring over a field $\mathbb{K}$, and let $I\subseteq S$ be a radical homogeneous ideal whose associated primes are generated by linear forms. Let $r_I$ denote the stabilization point of the minimum distance function $\delta_I$ of $I$, and let $\operatorname{reg}(S/I)$ denote the Castelnuovo–Mumford regularity of $S/I$.

Regularity bound conjecture. One should have

$$r_I\leq \operatorname{reg}(S/I).$$

The bound would extend the known inequality for ideals with $\dim(S/I)=1$ to the stated class of radical homogeneous ideals. However, the conjecture does not hold in general.