Partial gamma-positivity conjecture for quasi-Stirling permutation polynomials

For a multiset $\mathcal{M}$, let $\overline{\mathcal{Q}}_{\mathcal{M}}$ denote the set of quasi-Stirling permutations of $\mathcal{M}$, and define

$$\overline{Q}_{\mathcal{M}}(x,y,z)=\sum_{\pi\in \overline{\mathcal{Q}}_{\mathcal{M}}}x^{asc(\pi)}y^{des(\pi)}z^{plat(\pi)},$$

where $asc(\pi)$, $des(\pi)$, and $plat(\pi)$ are respectively the numbers of ascents, descents, and plateaux of $\pi$. Partial gamma-positivity conjecture. For every multiset $\mathcal{M}$, the polynomial $\overline{Q}_{\mathcal{M}}(x,y,z)$ is partial $\gamma$-positive. Lin, Ma, and Zhang proved the analogous assertion for $Q_{\mathcal{M}}(x,y,z)$, the polynomial for Stirling multipermutations, but the quasi-Stirling version remains the conjectural extension posed in that work.