The positivity conjecture for the Macdonald character operator

Let $p_\mu\left[X\frac{1-t}{1-q}\right]$ denote the indicated power-sum basis element, and let $q'=\frac{1-q}{1-t}$, with parameters $q',\gamma,v,z$. For partitions $\mu$ and $\nu$ satisfying $|\nu|-|\mu|=n$, consider the scalar product coefficient

$$\left\langle[z^n]\boldsymbol{\Gamma}(z,v)\cdot p_\mu\left[X\frac{1-t}{1-q}\right],p_\nu\left[X\frac{1-q}{1-t}\right]\right\rangle.$$

The positivity conjecture. The quantity

$$(-1)^nt^{|\mu|}\left\langle[z^n]\boldsymbol{\Gamma}(z,v)\cdot p_\mu\left[X\frac{1-t}{1-q}\right],p_\nu\left[X\frac{1-q}{1-t}\right]\right\rangle$$

is a polynomial in $-v,q',\gamma$ with non-negative integer coefficients. This conjecture proposes positivity for the operator's matrix coefficients in the stated basis; the supplied text gives no resolution evidence.