Bergeron–Garsia–Haiman–Tesler Conjecture II on nabla and modified Hall–Littlewood polynomials

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t]$ be the modified Hall–Littlewood polynomial obtained by specializing the modified Macdonald polynomial at $q=0$, and let $\nabla$ denote the nabla operator. Write $\ell(\mu)$ for the number of parts of $\mu$ and take the scalar product with the Schur function $s_{\lambda}$. Bergeron–Garsia–Haiman–Tesler's Conjecture II. For any partitions $\lambda$ and $\mu$,

$$\left\langle(-1)^{|\mu|-\ell(\mu)}\nabla \widetilde{H}_{\mu}[X;0,t],s_{\lambda}\right\rangle\in\mathbb{N}[q,t].$$

This asserts Schur positivity for the specified signed nabla transform. The conjecture was posed by Bergeron, Garsia, Haiman, and Tesler; the supplied source does not establish whether it has been resolved.