The touchpoint 4-variable Catalan conjecture

Let $\operatorname{Cat}_{n,r}(q,t,z,w)$ and $\operatorname{Cat}'_{n,r}(q,t,z,w)$ be the refinements of the 4-variable Catalan polynomials obtained by summing over Dyck paths of order $n$ with exactly $r$ rows of area zero, with the rise and valley products defined in the source. Let $E_{n,r}$ be the polynomials defined in the cited Schröder-path reference, and let $\left.f(z,w)\right|_{z^kw^\ell}$ denote coefficient extraction. Touchpoint 4-variable Catalan conjecture. For integers $n\geq k$, $\ell,r\geq0$,

$$\left.\operatorname{Cat}_{n,r}(q,t,z,w)\right|_{z^kw^\ell}=\left.\operatorname{Cat}'_{n,r}(q,t,z,w)\right|_{z^kw^\ell}$$ $$=\left\langle\Delta_{h_\ell}\nabla E_{n-\ell,r},s_{k+1,1^{n-k-\ell-1}}\right\rangle$$ $$=\left\langle\Delta_{h_\ell}\Delta'_{e_{n-k-\ell-1}}E_{n-\ell,r},e_{n-\ell}\right\rangle.$$

The source notes that the polynomials $E_{n,r}$ are defined externally and that the equalities refine the 4-variable Catalan conjecture according to returns to the diagonal; it does not provide a resolution.