Maximal Hilbert function conjecture for superfat points

Let $m,n\geq 2$. An $m$-superfat point is a zero-dimensional scheme in projective space with the superfat-point structure described in the source.

Maximal Hilbert function conjecture. For every $m,n\geq 2$, there exists an $m$-superfat point in $\mathbb{P}^2$ having maximal Hilbert function.

The surrounding examples show that superfat points in the plane need not have the same form as complete intersections. The source gives no proof or resolution of this existence claim; its quantifier includes $n$, although the ambient space in the assertion is $\mathbb{P}^2$.