Modified Segre conjecture for fat points in arbitrary position

Fix integers $d\ge 2$ and $n>2$. Let $S\subseteq {\mathbb P}^n$ be a finite collection of $s$ points, fix integers $m_1,\dots,m_s$, and set

$$Z:=\sum_{i=1}^s m_i p_i.$$

Let $\operatorname{w}(Z)$ be the total weight of $Z$, and for a linear subspace $L\subseteq {\mathbb P}^n$ let $\operatorname{w}_L(Z)$ be the sum of the multiplicities of the points lying in $L$. Modified Segre conjecture. Then $h^1(\mathcal I_Z(d))=0$ if all the following conditions hold: $\operatorname{w}(Z)\le nd+1$; $\operatorname{w}_L(Z)\le d+1$ for each line $L\subseteq {\mathbb P}^n$; and, for every integer $r=2,\dots,n-1$ and every $r$-dimensional linear subspace $L\subseteq {\mathbb P}^n$, $\operatorname{w}_L(Z)\le rd+2$, with the additional assumption $h^1(L,\mathcal I_{Z\cap L}(d))=0$ whenever $\operatorname{w}_L(Z)=rd+2$. This conjecture modifies Segre's conjecture for points in arbitrary position by imposing weight conditions on linear subspaces and a boundary cohomology condition. The supplied text presents examples and evidence but no resolution, so the conjecture remains open.