Projective-dimension 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 $\operatorname{bight}(J_t(T))$ denote the big height of this ideal. Projective-dimension conjecture for t-connected ideals. One has

$$\operatorname{pd}(R/J_t(T))=\operatorname{bight}(J_t(T)).$$

This conjecture is proposed alongside the regularity formula for $t$-connected ideals because these ideals may provide a more suitable generalization of edge ideals for homological invariants; the supplied text gives no resolution.