The superpolynomial lattice-path formula

Let $m$ and $n$ be coprime positive integers, let $\mathcal{P}_{n,m}(u,q,t)$ denote the superpolynomial, and let $D$ range over lattice paths below the main diagonal in an $m\times n$ rectangle. For a vertex $P$ of $D$, write $v(D)$ for the vertex set, and let $\delta_{m,n}=\frac{(m-1)(n-1)}{2}$; let $h_{+}(D)$ and $\beta(P)$ be the stated combinatorial statistics. Superpolynomial lattice-path conjecture. One has

$$\mathcal{P}_{n,m}(u,q,t)=\sum_D q^{\delta_{m,n}-|D|}t^{-h_{+}(D)}\prod_{P\in v(D)}(1-ut^{\beta(P)}).$$

For $m=n+1$ this was proved, and the identity was proved in the specialization $t=1$; it has also been verified computationally for many values of $m$ and $n$.