Harbourne's symbolic-power containment conjecture

Let $\mathbb{k}$ be a field, let $R=\mathbb{k}[x_1,\ldots,x_d]$, and let $I\subseteq R$ be a radical homogeneous ideal of bigheight $h$. Harbourne's conjecture. One has

$$I^{(h(n-1)+1)}\subseteq I^n$$

for every $n\geqslant 1$. This refines the uniform symbolic-power containment in prime characteristic and asks for a bound valid for radical homogeneous ideals in polynomial rings. The conjecture is false in general.