Circular LLT power-sum positivity conjecture

Let ${\boldsymbol\nu}$ determine a circular vertical strip digraph. Circular LLT power-sum positivity conjecture. Then $\omega \mathrm{G}_{\boldsymbol\nu}(\mathbf{x};q+1)$ is $\mathrm{p}$-positive. Furthermore, if

$$\omega \mathrm{G}_{\boldsymbol\nu}(\mathbf{x};q+1)=\sum_{\lambda}c_{{\boldsymbol\nu},\lambda}(q)\frac{\mathrm{p}_\lambda(\mathbf{x})}{z_\lambda},$$

the coefficients $c_{{\boldsymbol\nu},\lambda}(q)$ are polynomials with unimodal and non-negative integer coefficients. This is proposed from computer experiments as a generalization of a known positivity result and remains open.