Integrality and positivity conjecture for Macdonald-type integral binomial coefficients

Let $\lambda$ and $\mu$ be partitions with $\lambda\supseteq\mu$, and let $B_{\lambda\mu}$ be the integral binomial coefficient for the interpolation-polynomial family $\mathcal F$. Let $\mathbb I^+$ denote the positivity semiring in the parametrization $(\gamma,\tau,\alpha)$, allowing the sign and powers of $q=1+\gamma$, $t=1+\gamma\tau$ and $a=1+\gamma\alpha$ specified in the source. Integrality and positivity conjecture. For the families $\mathcal F={A\mathrm{M}}$ and $\mathcal F={B\mathrm{M}}$, the coefficient $B_{\lambda\mu}$ lies in $\mathbb I^+$ in that sense. The adjacent case is established, whereas the general binomial-coefficient assertion is presented as open.