Higher-harmonic Frobenius characteristic conjecture for the trivariate specialization

Let $\mathscr{H}_n^{(r)}(\mathbf{w};q,1,1)$ denote the graded Frobenius characteristic of the trivariate diagonal higher-harmonic space, with $\Dyck{r}{n}$ the set of $r$-Dyck paths, $\chi(\alpha\leq\beta)$ the indicator of the $r$-Tamari order relation, $d(\alpha,\beta)$ the associated distance statistic, and $e_{\gamma(\beta)}(\mathbf{w})$ the elementary symmetric function indexed by the composition $\gamma(\beta)$. The higher-harmonic Frobenius characteristic conjecture.

$$\mathscr{H}_n^{(r)}(\mathbf{w};q,1,1)=\sum_{\alpha,\beta\in\Dyck{r}{n}}\chi(\alpha\leq\beta)\,q^{d(\alpha,\beta)}\,e_{\gamma(\beta)}(\mathbf{w}).$$

This conjecture gives an explicit $r$-Dyck-path formula for the graded ${\mathbb S}_n$-character of the trivariate higher-harmonic space; its status is not determined in the supplied source context.