The equivariant Schur Lam–Postnikov–Pylyavskyy inequality

Let $\lambda$ and $\mu$ be partitions, let $\mathbf x_k=(x_1,\ldots,x_k)$, and let $s_\lambda(\mathbf x_k;\mathbf y)$ denote the double Schur polynomial. Write $\lambda\vee\mu$ and $\lambda\wedge\mu$ for the join and meet of the two partitions, and interpret $\leq_s^{\mathrm{equiv}}$ as the equivariant Schur-positivity order. Equivariant Schur LPP inequality.

$$s_\lambda(\mathbf x_k; \mathbf y)s_\mu(\mathbf x_k; \mathbf y) \leq_s^{\mathrm{equiv}} s_{\lambda\vee\mu}(\mathbf x_k; \mathbf y)s_{\lambda\wedge\mu}(\mathbf x_k; \mathbf y).$$

At $\mathbf y=\mathbf0$, this specializes to the classical LPP inequality. It has been checked for all Grassmannians in $S_n$ with $n\leq7$ and randomly for Grassmannians in $S_{10}$, but remains open in general.