Iarrobino's generalised Nagata conjecture for equal fat points

Let $(r,d)$ be integers satisfying

$$d\geq2,\qquad r\geq\max(d+5,2^d),$$

and $(r,d)\notin\{(7,2),(8,2),(9,3)\}$. Let $l(d,\delta,\mu^r)$ denote the dimension of the space of homogeneous degree-$\delta$ polynomials on ${\mathbb P}^d$ vanishing to order $\mu$ at $r$ general points. Iarrobino's generalised Nagata conjecture. If

$$\delta<\sqrt[d]{r}\,\mu,$$

then $l(d,\delta,\mu^r)=0$. This extends Nagata's critical-degree prediction from the plane to higher-dimensional projective spaces. The source presents it as Iarrobino's conjecture and does not state a resolution.