Standard atom positivity conjecture for flagged LLT polynomials

Let ${\boldsymbol\nu}$ be a tuple of ordinary, non-ragged-right skew diagrams, and let $\sigma$ be the standard compatible permutation. Let $\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]$ be the associated flagged LLT polynomial. Standard atom positivity conjecture. The polynomial

$$\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]$$

is atom positive, meaning that all of its Demazure-atom coefficients lie in $\mathbb N[t]$.

This conjecture generalizes the known Schur positivity of symmetric LLT polynomials. The source gives no resolution, so it remains open.