Alexandersson–Panova unimodality conjecture for LLT polynomial coefficients

Let $G$ be an oriented graph with no loops or multiple edges. Write the relevant LLT polynomial expansion as

$$\omega\mathrm{G}_G(\mathbf{x};q+1)=\sum_{\alpha}c^G_{\alpha}(q)\frac{\Psi_{\alpha}(\mathbf{x})}{z_{\alpha}},$$

where $\Psi_{\alpha}$ is the quasisymmetric power sum indexed by the composition $\alpha$, $z_{\alpha}$ is its standard normalization, and $c^G_{\alpha}(q)\in\mathbb{N}[q]$. Alexandersson–Panova conjecture. The polynomials $c^G_{\alpha}(q)$ are unimodal for all compositions $\alpha$. The source attributes the conjecture to P. Alexandersson and G. Panova and notes that computer experiments suggest an extension to the more general vertical-strip setting; the supplied text gives no resolution evidence.