Gao's beta-Grothendieck positivity conjecture

For $w\in S_n$, let $\mathfrak G^{(\beta)}_w(x_1,\dots,x_{n-1})$ be the $\beta$-Grothendieck polynomial, set $\nu^{(\beta)}_w=\mathfrak G^{(\beta)}_w(1,\dots,1)$, and define $c^{(\beta)}_w\in\mathbb Z[\beta]$ recursively by $c^{(\beta)}_\emptyset=1$ and

$$c^{(\beta)}_w=\nu^{(\beta)}_w-\sum_{u<w}c^{(\beta)}_up_u(w).$$

Here $p_u(w)$ is the number of occurrences of the permutation pattern $u$ in $w$.

Gao's beta-Grothendieck conjecture. For all permutations $w$, $c^{(\beta)}_w\in\mathbb Z_{\geq0}[\beta]$.

The conjecture is an analogue of Gao's positivity conjecture for principal specializations of Schubert polynomials. The coefficients have been confirmed computationally to be nonnegative for permutations of size at most $9$, and the paper proves the claim for vexillary $1243$-avoiding permutations; the general case remains open.