Beck–Brauner–? conjecture for $q,t$-symmetry of $(a,b,\ldots,b)$-Dyck paths

Let $\vec{k}$ be a vector of positive integers and define the area-bounce polynomial

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

A polynomial is $q,t$-symmetric when $F(q,t)=F(t,q)$. The $(a,b,\ldots,b)$-symmetry conjecture. For any positive integers $a$ and $b$, if

$$\vec{k}=(a,b,b,\ldots,b),$$

then

$$C_{\vec{k}}(q,t)=C_{\vec{k}}(t,q).$$

The conjecture is attributed in the paper to the authors of the cited polyhedral-geometry work. The cited work proposes this family, while the supplied excerpt gives no proof or later resolution.