Stable atom positivity conjectures for nonsymmetric LLT expressions

Let stable atoms be the basis elements ${\mathcal A}_{\eta|\lambda}$ defined from Demazure atoms, and say that an element is stable atom positive when it is a nonnegative linear combination of stable atoms. Let ${\mathsf \Pi}_\ell$, ${\mathsf{M}}^{\mathsf{Q}}_{\alpha,k}$, $\underline{\tau_{qtu,\ell}^*}$, $\boldsymbol{\nabla}$, and ${\mathsf{C}}_\alpha$ denote the operators and quantities defined in the paper. The stable atom positivity conjectures.

1. Flagged row and column LLT polynomials are stable atom positive.
2. The quantity ${\mathsf \Pi}_\ell\,{{\mathsf{M}}^{\mathsf{Q}}_{\alpha,k}}$ appearing in the paper's equation (1) is stable atom positive.
3. The quantity

$$(-1)^{n+k}\left[\underline{\tau_{qtu,\ell}^*}^{-1}\boldsymbol{\nabla}^{-1}\underline{\tau_{qtu,\ell}^*}^{-1}\right]_k{\mathsf{C}}_\alpha$$

appearing in the paper's equation (2) is stable atom positive.

These are proposed analogues of stable atom positivity results for nonsymmetric shuffle and flagged LLT settings. The source presents them as conjectures and gives no resolution, so their status remains open.