The $(q,t)$-deformed hook product formula for connected $d$-complete posets

Let $P$ be a connected $d$-complete poset with maximum element $v_0$, rank function $r:P\to\mathbb{N}$, top tree $T$, and $d$-complete coloring $c:P\to T$. For a $P$-partition $\sigma\in\mathcal{A}(P)$, let $W_P(\sigma;q,t)$ be the weight defined above, and let $\boldsymbol{z}^\sigma$ and $\boldsymbol{z}[H_P(v)]$ use the notation introduced in the paper. The $(q,t)$-deformed hook product formula. For every connected $d$-complete poset $P$,

$$\sum_{\sigma\in\mathcal{A}(P)}W_P(\sigma;q,t)\boldsymbol{z}^\sigma=\prod_{v\in P}F(\boldsymbol{z}[H_P(v)];q,t).$$

This is proposed as a $(q,t)$-deformation of the Peterson--Proctor hook formula. The supplied text does not state whether the conjecture has been proved or disproved.