Bergeron–Garsia–Haiman–Tesler signed Schur-positivity conjecture for nabla of monomial symmetric functions

Let $n$ be a positive integer, and let $mu$ and $\lambda$ be partitions of $n$. Let $m_\mu$ be the monomial symmetric function, $s_\lambda$ the Schur function, and $\nabla$ the Bergeron–Garsia nabla operator on symmetric functions. Bergeron–Garsia–Haiman–Tesler's conjecture. For any partitions $\mu,\lambda\vdash n$,

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

The paper proves this conjecture, together with the stronger statement obtained by replacing $\nabla$ with $\nabla^r$ for every integer $r\geq 1$.