Marked-Dyck-path formula at $q=1$ for torus-knot homology

Let $\mathcal{P}_{s}^{2k}(T_{n,m})(q,t)$ denote the relevant coefficient of the triply graded Poincaré polynomial, and let the sum range over all marked Dyck paths in the $m\times n$ rectangle with $k$ marks; write $S(\pi)$ for the area above the path. Marked-Dyck-path conjecture.

$$\mathcal{P}_{s}^{2k}(T_{n,m})(1,t)=t^{k}\sum_{\pi}t^{2S(\pi)}.$$

This extends the preceding $(n,n+1)$ formula to general torus knots and gives a combinatorial model for the $q=1$ specialization; the source does not provide a resolution status.