Rational shuffle Frobenius-character conjecture

Let $\operatorname{Fr}_{n,m}$ be the symmetric function obtained by summing over $m/n$ parking functions with the statistics and Gessel quasisymmetric functions defined in the source, and let $\widetilde{P}_{n,m}$ be the transformed DAHA element. Rational shuffle Frobenius-character conjecture.

$$\operatorname{Fr}_{n,m}=\widetilde{P}_{n,m}\cdot1.$$

For $m=n+1$, the corresponding specialization was proved in the cited work; the general rational identity is presented as an open conjecture and is supported by the geometric interpretation of $\operatorname{Fr}_{n,m}$.