The regularity-index conjecture for ideals with linear associated primes

Let $S$ be the standard graded polynomial ring in the paper, and let $I\subset S$ be an unmixed radical graded ideal. Assume that every associated prime of $I$ is generated by linear forms. Let $r_0$ be the regularity index of the minimum distance function $\delta_I$, namely the least integer at which $\delta_I$ reaches and thereafter remains equal to $1$. The regularity-index conjecture. One has

$$\delta_I(d)=1\quad\text{for all }d\geq\operatorname{reg}(S/I),$$

that is, $r_0\leq\operatorname{reg}(S/I)$. This would extend the known bound for graded vanishing ideals of finite sets of projective points over finite fields. The general case remains open, while the regularity index is described as difficult to compute.