The generic Hilbert function conjecture for general unions of flat fat points

Let $Z_{s,m}$ be a general union of $s$ flat fat points of multiplicity $m$ in $\mathbb P^2$. Assume $m\geq 4$ and $s\geq 2m-3$. Its Hilbert function should satisfy

$$h_{Z_{s,m}}(j)=\min\left\{ms,\binom{j+2}{2}\right\}.$$

A flat fat point is the specialization of a fat point supported on a line, and “general union” means that the supports are chosen generally. This conjecture would extend the technical generic-Hilbert-function result used in the paper from lower multiplicities to all $m\geq4$ under the stated numerical condition.