Ribbon-tableau positivity conjecture for H-functions

Let $H^{(k)}_\mu(X;q)$ be the symmetric functions defined from $k$-ribbon tableaux, where $k$ is the level, and expand each $H^{(k)}_\mu$ in the Schur basis.

Positivity conjecture. The coefficients in these Schur-basis expansions are polynomials with nonnegative integer coefficients.

The conjecture proposes a Schur-positivity property for the ribbon-tableau $H$-functions; the source presents it as supported by strong experimental evidence, with no proof supplied there.