Integrality and positivity conjecture for integral binomial coefficients of interpolation polynomials

Let $\lambda$ and $\mu$ be partitions with $\lambda\supseteq\mu$, and let $B_{\lambda\mu}$ denote the integral binomial coefficient for the interpolation-polynomial family $\mathcal F$. Let $\mathbb I^+$ denote the semiring of polynomials with non-negative integer coefficients in the relevant parameter(s). Integrality and positivity conjecture. For the families $\mathcal F={A\mathrm{J}}$ and $\mathcal F={B\mathrm{J}}$, the coefficient $B_{\lambda\mu}$ is a polynomial with non-negative integral coefficients in the parameter(s), so that $B_{\lambda\mu}\in\mathbb I^+$. The adjacent coefficients are known to have this property, while the corresponding assertion for general binomial coefficients remains open.