Positivity of generalized Q-Kostka polynomials

Let $\mu^*=(\mu^{(1)},\mu^{(2)},\ldots,\mu^{(k)})$ be a sequence of partitions, let $\bar{\mu^*}$ be their concatenation, and define the generalized $Q$-Kostka polynomials by

$$G_{\mu^*}[X;q]=\sum_{\lambda}L_{\lambda;\mu^*}(q)Q_{\lambda}[X].$$

Generalized Q-Kostka positivity conjecture. If $\bar{\mu^*}$ is a partition, then $L_{\lambda;\mu^*}(q)$ is a polynomial in $q$ with non-negative integer coefficients. These coefficients are proposed as $q$-analogs of structure coefficients for products of Schur $Q$-functions, paralleling the role of parabolic Kostka polynomials; the paper gives computational motivation but no general proof.