Monomial Harbourne–Hirschowitz conjecture

Let $S=\mathbb P^2$, let $I=(y,x^r)^m$ with $r$ and $m$ natural numbers and $r>9$, and let $Y$ be a general member of ${\rm Hilb}_I S$. For a divisor $D=dH$ on $\mathbb P^2$, the monomial Harbourne–Hirschowitz conjecture asserts that, for every $d>0$,

$$H^0(\mathcal I_Y\otimes\mathcal O_{\mathbb P^2}(D))=\max\left\{0,\binom{d+2}{2}-r\binom{m+1}{2}\right\}.$$

This is presented as a monomial analogue related to the Nagata and SHGH conjectures, and it implies the uniform Harbourne–Hirschowitz conjecture. Its resolution is not stated in the source.