Maximal Hilbert function conjecture for generic 2-square points

Let $Z_s$ be a scheme made of $s$ generic 2-square points, where the support points are in generic position and the $2s$ lines defining their ideals are also in generic position. Let $H(Z_s,t)$ denote its Hilbert function.

Maximal Hilbert function conjecture. For every $s$ and $t$,

$$H(Z_s,t)=\min\left\{4s,\binom{t+2}{2}\right\}.$$

The claim extends the computation obtained by the Horace method for $s\leq 6$, asserting that generic 2-square points always have the Hilbert function expected from $4s$ generic points. The source gives no resolution beyond those computed cases.