The vexillary equivariant Schubert 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 $\mathfrak S_w(\mathbf x;\mathbf y)$ be the double Schubert polynomial, and let $\alpha_i=y_{i+1}-y_i$. Define $f\leq_{\mathfrak S}^{\mathrm{equiv}}g$ by requiring that $g-f$ be a nonnegative polynomial combination of double Schubert polynomials with coefficients in $\mathbb R_{\geq0}[\alpha_1,\alpha_2,\ldots]$. Let $u\vee v$ and $u\wedge v$ denote the relevant join and meet. Vexillary equivariant Schubert LPP inequality. If $\phi(u)\preceq\phi(v)$, then

$$\mathfrak S_u(\mathbf x; \mathbf y) \mathfrak S_v(\mathbf x; \mathbf y) \leq_{\mathfrak S}^{\mathrm{equiv}} \mathfrak S_{u\vee v}(\mathbf x; \mathbf y) \mathfrak S_{u\wedge v}(\mathbf x; \mathbf y).$$

Setting $\mathbf y=\mathbf0$ gives the classical vexillary Schubert inequality. The conjecture was tested randomly for vexillary permutations in $S_9$, but remains open in general.