Fatabbi–Lorenzini's generalized Segre bound for fat points

Let $Z$ be a collection of fat points in ${\mathbb P}^n$. For a linear $r$-subspace $L\subseteq {\mathbb P}^n$, let $\operatorname{w}_L(Z)$ be the sum of the multiplicities of the points of $Z$ lying in $L$, and let $\operatorname{reg}(Z)$ denote its regularity index. For $r=1,\dots,n$ and for any linear $r$-subspace $L$ of ${\mathbb P}^n$, set

$$T(Z,L):=\left\lfloor\frac{\operatorname{w}_L(Z)+r-2}{r}\right\rfloor.$$

Fatabbi–Lorenzini's generalized Segre bound. One has

$$\operatorname{reg}(Z)\le\max\{T(Z,L): L\subseteq {\mathbb P}^n\}.$$

This conjecture seeks an upper bound for the regularity index of arbitrary fat points, extending Segre's bound beyond points in linearly general position. The supplied text gives no resolution evidence, so its status remains open.