Nonnegative coefficients of the Q-Kostka polynomials

Let $L_{\lambda\mu}(q)$ denote the $Q$-Kostka polynomial, with $\lambda$ and $\mu$ partitions in the indexing convention of the paper. Positivity conjecture. The $Q$-Kostka polynomials $L_{\lambda\mu}(q)$ have non-negative coefficients. This is presented as the property most suggestive of an analogy with the Kostka–Foulkes polynomials. The paper proves the claim for some particular cases and proposes a combinatorial interpretation via a statistic on marked shifted tableaux, but leaves the general statement open.