The symmetric theta conjecture for the nabla-star operator

Let $n$ be a positive integer, let $\nabla_\ast$ be the operator on symmetric functions in two alphabets defined by $\nabla_\ast\widetilde H_\mu=\widetilde H_\mu\otimes\widetilde H_\mu$, and let $\mathsf{RTT}_0(n)$ denote the relevant family of rooted tiered trees of size $n$. For each $T$, write $\mathsf{inv}(T)$ for its inversion statistic and $\alpha(T),\beta(T)$ for the associated compositions. The symmetric theta conjecture.

$$\left.\nabla_\ast e_n\right\rvert_{t=1}=\sum_{T\in\mathsf{RTT}_0(n)}q^{\mathsf{inv}(T)}x^{\alpha(T)}y^{\beta(T)}.$$

This is the equivalent symmetric-function reformulation obtained by summing the preceding identity over compositions; the supplied text gives no resolution status.