SHGH conjecture for equal multiplicities at general points

Let $Z=p_1+cdots+p_s eq\mathbb P^2$ be a union of general points, where either $s$ is a square or $s>8$, and let $I(mZ)$ denote the degree-$m$ fat-point ideal. SHGH conjecture. The dimension of its degree-$t$ piece is

$$\dim [I(mZ)]_t=\max\left\{0,\binom{t+2}{2}-s\binom{m+1}{2}\right\}.$$

This is presented as a special case of the Segre–Harbourne–Gimigliano–Hirschowitz conjecture, which predicts when general fat points impose independent conditions; the stated case remains conjectural.