Bivariate combinatorial interpretation of the q,p-Catalan triangle

Let $C_{n,k}(q,p)$ be the $q,p$-Catalan triangle, and let $\mathfrak{S}'_{n,k}(312)$ denote the set of $312$-avoiding permutations represented in the triangle. For a permutation $\pi$, write $\operatorname{inv}(\pi)$ and $\operatorname{coinv}(\pi)$ for its inversion and coinversion statistics.

Bivariate combinatorial interpretation. For all $n\ge1$ and $0\le k\le n$:

$$C_{n,k}(q,p)=\sum_{\pi\in\mathfrak{S}'_{n,k}(312)}q^{\operatorname{inv}(\pi)}p^{\operatorname{coinv}(\pi)}.$$

The identity has been verified for $n\le5$; the source attributes a proof to W5. Its status should be checked against that reference.