Pun's Demazure-character atom-positivity conjecture

Let $\alpha,\beta\in\mathbb{N}^m$, let $\kappa_\alpha$ and $\kappa_\beta$ be Demazure characters, and let $\mathcal{A}_\gamma$ denote the Demazure atom indexed by $\gamma$. Pun's conjecture. There exist nonnegative integers $a_{\beta,\alpha}^{\gamma}$ such that

$$\kappa_{\beta}\kappa_{\alpha}=\sum_{\gamma}a_{\beta,\alpha}^{\gamma}\mathcal{A}_{\gamma}.$$

This is weaker than Polo's Schubert-character conjecture because Schubert characters expand nonnegatively into Demazure atoms. Pun proves the claim when $\alpha$ and $\beta$ have length at most $3$ and at most $2$ nonzero parts, but the general statement remains open.