Stable atom positivity conjectures for flagged LLT and modified nonsymmetric polynomials

Let $\mathcal P(r)$ be the polynomial space in which the stable atoms form a basis, and let stable atom positivity mean that the coefficients in this basis belong to $\mathbb N[q,t]$. Let $\mathcal G_{r,\boldsymbol\nu}(\mathbf x;t)$ be the flagged LLT polynomials, let $\mathsf G_r(\pi,\Sigma)(\mathbf x;t)$ and $\widetilde{\mathsf G}_r(\pi,\Sigma)(\mathbf x;t)$ be the flagged and column LLT polynomials, let $\check{\mathsf H}_{\eta|\lambda}$ be the modified $r$-nonsymmetric Macdonald polynomials, and let $\boldsymbol\nabla^{-1}\mathsf C_\alpha$ and $\mathsf\Pi_r\mathsf C_{m,n}^\alpha$ denote the quantities specified in the cited theorems. Stable atom positivity conjecture. The following elements are stable atom positive: (i) $\mathcal G_{r,\boldsymbol\nu}(\mathbf x;t)$, including $\mathsf G_r(\pi,\Sigma)(\mathbf x;t)$ and $\widetilde{\mathsf G}_r(\pi,\Sigma)(\mathbf x;t)$; (ii) $\check{\mathsf H}_{\eta|\lambda}$; (iii) $\boldsymbol\nabla^{-1}\mathsf C_\alpha$; and (iv) $\mathsf\Pi_r\mathsf C_{m,n}^\alpha$. Stable atom positivity strengthens Schur positivity under full Weyl symmetrization. The text notes that (i) implies (ii)--(iv), while recording them separately because each may have distinct representation-theoretic significance; no resolution is supplied.