Bergeron–Garsia–Haiman–Tesler Conjecture III on nabla, omega, and Hall–Littlewood polynomials

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t^{-1}]$ be the modified Hall–Littlewood polynomial specialized at $q=0$, let $\nabla$ be the nabla operator, and let $\omega$ be the standard involution on symmetric functions. A symmetric function is Schur positive when its Schur expansion has coefficients in $\mathbb{N}[t]$. Bergeron–Garsia–Haiman–Tesler's Conjecture III. For any partition $\mu$, $\nabla\omega\widetilde{H}_{\mu}[X;0,t^{-1}]$ is Schur positive. Moreover, if $\mu\unrhd\lambda$ in dominance order, then

$$\nabla\omega\widetilde{H}_{\mu}[X;0,t^{-1}]-\nabla\omega\widetilde{H}_{\lambda}[X;0,t^{-1}]$$

is also Schur positive. The conjecture was posed by Bergeron, Garsia, Haiman, and Tesler; the supplied source does not establish whether it has been resolved.