Bergeron–Garsia Schur-positivity conjecture for nabla applied to monomial symmetric functions

Let $\Lambda$ be the ring of symmetric functions, let $m_{\mu}$ and $s_{\lambda}$ denote the monomial and Schur symmetric functions indexed by partitions $\mu$ and $\lambda$, respectively, and let $\langle\,\cdot\,,\,\cdot\,\rangle$ be the Hall inner product. Let $|\mu|$ be the size of $\mu$, $l(\mu)$ its length, and let $\nabla$ be the modified-Macdonald eigenoperator. Bergeron–Garsia's conjecture. For every pair of partitions $\lambda$ and $\mu$,

$$\left\langle(-1)^{|\mu|-l(\mu)}\nabla m_{\mu},s_{\lambda}\right\rangle\in\mathbb{N}[q,t].$$

The claim asserts Schur positivity of the signed transformed monomial symmetric function. The paper notes known special cases, including $\mu=(1^n)$, $\mu=(n)$, and hook cases, while the general statement is not resolved in the supplied context.