Le Mogne–Préville-Ratelle's symmetry and Schur-positivity conjecture for ν-Tamari intervals

Let $\lambda$ be a triangular partition, and let its top-down tableau be the Young tableau associated with a maximal chain in the $\nu$-Tamari lattice. Say that this tableau is sim-sym when it has the similarity-symmetry property defined in the source. Le Mogne–Préville-Ratelle's conjecture. If the top-down tableau of $\lambda$ is sim-sym, then the $q,t$-enumeration of $\nu$-Tamari intervals is symmetric and Schur-positive.

This conjecture generalizes the preceding theorem for triangular $2$-partitions. The source explicitly presents it as a new conjecture, and notes that the general $q,t$-Catalan setting has many open questions.