The stable Harbourne conjecture for radical ideals

Let $I$ be a radical ideal of big height $c$ in a regular ring. For each positive integer $n$, let $I^{(n)}$ denote the $n$th symbolic power of $I$. Stable Harbourne conjecture. One has

$$I^{(cn-c+1)}\subseteq I^n\quad\text{for }n\gg 0.$$

The conjecture is a stable version of the Harbourne conjecture, concerning eventual containments between symbolic and ordinary powers. The paper's abstract states that this conjecture holds for ideals defining space monomial curves, while the supplied material does not establish its status in the stated generality.