Refined $q,t$-symmetry conjecture for $\vec{k}$-Dyck paths with $a$- and $b$-tails

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ denote the corresponding parameter indexing the relevant $\vec{k}$-Dyck paths, and let $\widetilde{C}_{\mathcal{K}^{a}\mathcal{K}^{b}}(q,t)$ be the refined polynomial graded by the statistics area and depth. A polynomial is $q,t$-symmetric when $F(q,t)=F(t,q)$. The $(a,b)$-tail symmetry conjecture. For every positive integer vector $\vec{k}$ and all positive integers $a$ and $b$,

$$\widetilde{C}_{\mathcal{K}^{a}\mathcal{K}^{b}}(q,t)=\widetilde{C}_{\mathcal{K}^{a}\mathcal{K}^{b}}(t,q).$$

The paper presents this as the second conjecture in its refined $q,t$-symmetry program. The nearby theorems prove symmetry for related refined polynomials, while the status of this stronger claim is left open in the supplied text.