Iarrobino's conjecture on Waldschmidt constants of very general points

Let $X$ be a very general set of $r$ points in $\mathbb{P}^N$ over a field of characteristic zero, with

$$r\geq \max\\{N+5,2^N\\},\qquad (r,N)\notin\\{(7,2),(8,2),(9,3)\\}.$$

Let $I(X)$ be the homogeneous ideal of $X$, let $\alpha(I(X)^{(n)})$ denote the least degree of a nonzero form in its $n$th symbolic power, and let $\widehat{\alpha}(I(X))$ denote its Waldschmidt constant.

Iarrobino's conjecture. For every $n\in\mathbb{N}$,

$$\alpha(I(X)^{(n)})\geq n\sqrt[N]{r}.$$

Equivalently, apart from the listed exceptions,

$$\widehat{\alpha}(I(X))=\sqrt[N]{r}.$$

The conjecture is known in the case $r=s^N$, but is open in general.