Hall–Littlewood positivity for q-divided symmetrized Schubert polynomials

Let $cmathfrak{S}_{w}cin coperatorname{Pol}_n$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. For a partition $clambda=(clambda_1cgeq \cdots cgeq clambda_n)$, let $P_{clambda}(x_1,\cdots,x_n;q^{-1})$ be the Hall–Littlewood $P$-polynomial with parameter $q^{-1}$. Hall–Littlewood expansion conjecture. For every Schubert polynomial $cmathfrak{S}_{w}cin coperatorname{Pol}_n$, there are Laurent polynomials $b_{clambda,w}(q)$ with nonnegative integer coefficients such that

$$clangle cmathfrak{S}_{w}crangle_n^q=csum_{clambda=(clambda_1cgeq \cdots cgeq clambda_n)}b_{clambda,w}(q)P_{clambda}(x_1,\cdots,x_n;q^{-1}).$$

This generalizes the earlier Hall–Littlewood positivity prediction and was verified in the source for all permutations up to $S_8$; the general case remains open.