Regularity conjecture for t-connected ideals of trees

Let $T$ be a tree, let $t$ be a positive integer, and let $J_t(T)$ denote the $t$-connected ideal of $T$. Let $\mathcal H$ be the hypergraph corresponding to $J_t(T)$, and write $\nu(\mathcal H)$ for its matching number. Regularity conjecture for t-connected ideals. One has

$$\operatorname{reg}(R/J_t(T))=(t-1)\nu(\mathcal H).$$

The conjecture is motivated by computational evidence and by the fact that the analogous formula holds in the examples discussed in the paper, but its general validity for trees is left open.