Grassmannian Billey–Postnikov decomposition conjecture for palindromic Schubert polynomials

Let $W$ be a Coxeter group, let $J$ be a subset of its simple generators, and let $w\in W^J$. Let $P_w^J$ denote the corresponding parabolic Kazhdan–Lusztig polynomial, and call it palindromic when its coefficients are symmetric. A Grassmannian Billey–Postnikov decomposition is a Billey–Postnikov decomposition relative to $J$ whose associated parabolic factor is Grassmannian. Grassmannian Billey–Postnikov decomposition conjecture. If $P_w^J$ is palindromic, then $w$ has a Grassmannian Billey–Postnikov decomposition. Consequently, the Ryan–Wolper theorem should hold in every Kac–Moody flag variety. This is motivated by results for finite type and for several classes of non-finite Weyl groups, but the source presents the assertion as conjectural and does not resolve it.