Stable Harbourne–Huneke conjecture for radical ideal containments

Let $I\subseteq k[x_0,\dots,x_n]$ be a homogeneous radical ideal of big height $h$, and let $\mathcal{M}=(x_0,\dots,x_n)$ be the graded maximal ideal. Stable Harbourne–Huneke conjecture. For all sufficiently large integers $r$, both containments hold:

$$I^{(hr)}\subseteq \mathcal{M}^{r(h-1)}I^r$$

and

$$I^{(hr-h+1)}\subseteq \mathcal{M}^{(r-1)(h-1)}I^r.$$

These are stable strengthened forms of symbolic-power containment conjectures; the supplied text gives no resolution status for either assertion.