Iarrobino's strengthened generalised Nagata conjecture for equal fat points

Let $(r,d)$ be integers satisfying

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

and $(r,d)\notin\{(7,2),(8,2),(9,2),(8,3),(9,3)\}$. 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 strengthened conjecture. If

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

then $l(d,\delta,\mu^r)=0$. This version strengthens the preceding formulation by including the equality case, while excluding the exceptional pairs that would contradict it. The source gives no evidence that the conjecture has been resolved.