Unimodal and anti-unimodal atom positivity conjecture for flagged LLT polynomials

Let ${\boldsymbol\nu},\sigma$ be any flagged LLT indexing data, and let $\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]$ be the associated flagged LLT polynomial. If

$$\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]=\sum_{\lambda}a_\lambda(t)\mathcal A_\lambda(x),$$

then $a_\lambda(t)=\langle\mathcal A_\lambda(x)\rangle\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]$. An index is unimodal if it is weakly increasing and then weakly decreasing, and anti-unimodal if it is weakly decreasing and then weakly increasing. Unimodal and anti-unimodal atom positivity conjecture. For every flagged LLT indexing datum, $a_\lambda(t)\in\mathbb N[t]$ whenever $\lambda$ is unimodal or anti-unimodal.

This conjecture gives another extension of symmetric LLT Schur positivity. The source provides no resolution, so it remains open.