Rational Catalan q,t-symmetry refinement

Let $Y_{m/n}$ be the set of $m/n$ Dyck paths, let $h_{+}(D)$ be the stated statistic, and define

$$C_{n,m}(q,t)=\sum_{D\in Y_{m/n}}q^{\delta_{m,n}-|D|}t^{h_{+}(D)},\qquad \delta_{m,n}=\frac{(m-1)(n-1)}{2}.$$

Let $\widetilde{P}_{n,m}$ be the transformed DAHA element, let $h_n$ be the complete symmetric function, and let $c_{n,m}(\lambda)$ and $g_\lambda$ be the coefficients defined in the source. Rational Catalan q,t-symmetry conjecture.

$$C_{n,m}(q,t)=(h_n\mid\widetilde{P}_{n,m}\mid1)=\sum_{\lambda\vdash n}\frac{c_{n,m}(\lambda)}{g_\lambda}.$$

Since the right-hand side is symmetric in $q$ and $t$, this would imply $C_{n,m}(q,t)=C_{n,m}(t,q)$; the source notes that the claim is known for $m=n+1$ and otherwise leaves it conjectural.