Huneke–Harbourne containment conjecture for fat point ideals

Let $Z\subseteq\mathbb P^N$ be a fat point subscheme, let $R=K[x_0,\ldots,x_N]$ be the homogeneous coordinate ring, and let $M=(x_0,\ldots,x_N)$. Huneke–Harbourne containment conjecture. For every integer $r\geq1$,

$$I(NrZ)\subseteq M^{Nr-r}I(Z)^r.$$

The conjecture was proposed as a strengthened containment motivated by optimality questions for symbolic-power containments; the source states that it remains open in all characteristics.