Alternant divisibility and Schur-positivity conjecture

Let $N$ be a positive integer, let $\mathcal{P}$ be the polynomial ring in variables $x_1,\ldots,x_N$, and for $\alpha=(\alpha_1,\ldots,\alpha_N)\in\mathbb{N}^N$ define

$$a_\alpha=\det\left((x_i^{\alpha_j})_{1\leq i,j\leq N}\right).$$

Let $\rho=(N-1,N-2,\ldots,0)$, so that

a_\rho=\prodnonlimits\limits_{1\leq i<j\leq N}(x_i-x_j).

Alternant divisibility and positivity conjecture. For every $\alpha\in\mathbb{N}^N$, the alternant $a_\alpha$ is a multiple of $a_\rho$ in $\mathcal{P}$. Furthermore, if $\alpha_1>\alpha_2>\cdots>\alpha_N$, then the polynomial $a_\alpha/a_\rho$ has positive, more precisely nonnegative, integer coefficients.

The divisibility follows from the alternating nature of $a_\alpha$ and the Vandermonde determinant, while the coefficient-positivity assertion is the substantive part, identifying these quotients with the usual Schur-polynomial phenomenon for strictly decreasing exponent sequences. The source presents the statement as a suspicion, and no resolution is supplied in the given text.