The vexillary Grothendieck Lam–Postnikov–Pylyavskyy inequality

Let $u,v\in S_n$ be vexillary permutations, let $\phi(w)$ be the Wachs flag of $w$, and write $a\preceq b$ when $a$ is a prefix of $b$. Let $\mathfrak G_w^{(\beta)}(\mathbf x)$ be the $\beta$-Grothendieck polynomial, and define $f\leq_{\mathfrak G^{(\beta)}}g$ when $g-f$ is a nonnegative polynomial combination of $\beta$-Grothendieck polynomials with coefficients in $\mathbb R_{\geq0}[\beta]$. Let $u\vee v$ and $u\wedge v$ denote the relevant join and meet. Vexillary Grothendieck LPP inequality. If $\phi(u)\preceq\phi(v)$, then

$$\mathfrak G_u^{(\beta)}(\mathbf x)\mathfrak G_v^{(\beta)}(\mathbf x) \leq_{\mathfrak G^{(\beta)}} \mathfrak G_{u\vee v}^{(\beta)}(\mathbf x)\mathfrak G_{u\wedge v}^{(\beta)}(\mathbf x).$$

At $\beta=0$, this specializes to the classical vexillary Schubert inequality. The conjecture has been checked for $n\leq7$ and remains open in general; its Grassmannian specialization was previously predicted by Thomas and Yong.