Trivariate Shuffle Conjecture for higher harmonics

Let $\mathscr{H}_n^{(r)}(\mathbf{w};q,t,1)$ denote the graded Frobenius characteristic of the trivariate diagonal higher-harmonic space. Let $\Dyck{r}{n}$ be the set of $r$-Dyck paths, let $\mathcal{W}_\beta$ be the set of $r$-semi-parking functions associated with $\beta$, let $\chi(\alpha\leq\beta)$ indicate the $r$-Tamari order relation, let $d(\alpha,\beta)$ be the associated distance statistic, let $\operatorname{dinv}_r(\varphi)$ be the refined diagonal-inversion statistic, and let $\mathbf{w}_\varphi$ be the monomial associated with $\varphi$. The trivariate Shuffle Conjecture.

$$\mathscr{H}_n^{(r)}(\mathbf{w};q,t,1)=\sum_{\alpha,\beta\in\Dyck{r}{n}}\sum_{\varphi\in\mathcal{W}_\beta}\chi(\alpha\leq\beta)\,q^{d(\alpha,\beta)}t^{\operatorname{dinv}_r(\varphi)}\mathbf{w}_\varphi.$$

This is proposed as a trivariate extension of the bivariate Shuffle Conjecture and supplies a combinatorial formula for the trivariate higher-harmonic Frobenius characteristic; the supplied source gives no resolution evidence.