Reiner–Shimozono's positivity conjecture for products of key polynomials

Let $a$ and $b$ be weak compositions, and let ${\mathfrak{D}}_a$ and ${\mathfrak{D}}_b$ denote the corresponding key polynomials. Let ${\mathfrak{A}}_c$ denote the Demazure atom indexed by a weak composition $c$. Reiner–Shimozono's conjecture. The product

$${\mathfrak{D}}_a\cdot {\mathfrak{D}}_b$$

expands positively in Demazure atoms, meaning that it is a sum of Demazure atoms with nonnegative integer coefficients. This conjecture concerns positivity in the Demazure atom basis despite the fact that neither key polynomials nor Demazure atoms have positive structure coefficients in general; a generalization is known in subsequent work, while the conjecture itself is presented here as unresolved.