Xin's partition-type $q,t$-symmetry conjecture for $\vec{k}$-Dyck paths

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ be the associated parameter, and define the area-bounce polynomial

$$C_{\mathcal{K}}(q,t)=\sum_{\pi\in\mathcal{D}_{\mathcal{K}}}q^{\mathrm{area}(\pi)}t^{\mathrm{bounce}(\pi)}.$$

Write $\lambda(\mathcal{K})$ for the partition associated with $\mathcal{K}$. A polynomial is $q,t$-symmetric when $F(q,t)=F(t,q)$. Xin's partition-type symmetry conjecture. For positive integers $a$, $s$, and $m$, if

$$\lambda(\mathcal{K})=((a+1)^s,a^m),$$

then

$$C_{\mathcal{K}}(q,t)=C_{\mathcal{K}}(t,q).$$

This conjecture extends the special partition families for which symmetry is known in the paper. It is attributed to Xin and is presented here as an experimental conjecture; no resolution is given in the supplied text.