Zabrocki's nonnegativity conjecture for set-partition sums

For a positive integer $m$, let $[m]=\{1,\dots,m\}$, and let $\sigma$ and $\tau$ be set partitions of $[m]$, with $\tau\ge\sigma$ meaning that every block of $\sigma$ is contained in some block of $\tau$. Write $\ell(\sigma)$ for the number of blocks of $\sigma$, let $(\lambda(\tau)-1)!$ denote the product of the factorials of one less than the number of elements in each block of $\tau$, and let $(\lambda(\sigma,\tau)-1)!$ denote the product of the factorials of one less than the number of blocks of $\sigma$ contained in each block of $\tau$. Zabrocki's conjecture. For every set partition $\sigma$ of $[m]$, the numbers

$$\sum_{\tau\ge\sigma}(-1)^{\ell(\sigma)-\ell(\tau)} \frac{(\ell(\tau)-1)!(\lambda(\sigma,\tau)-1)!}{(\lambda(\tau)-1)!}$$

are nonnegative. This conjecture concerns the sign pattern in the expansion of the $\mathbf{x}$-basis in the $\mathbf{e}$-basis for symmetric functions in noncommuting variables. Its resolution is not specified in the supplied text.