Relative Hodge–Riemann conjecture for Soergel bimodule products

Let $W$ be the Weyl group under consideration, let $x_1,\dots,x_m\in W$, and let $B_{x_1}\dots B_{x_m}$ be equipped with its intersection form. For $\mathbf a=(a_1,\dots,a_{m-1})\in\mathbb R_{>0}^{m-1}$, let $L_{\mathbf a}$ be the associated Lefschetz operator, and write $RHR(x_1,\dots,x_m)$ for the assertion that $L_{\mathbf a}$ satisfies relative Hodge–Riemann for every such $\mathbf a$. Relative Hodge–Riemann conjecture. For any $x_1,\dots,x_m\in W$, $RHR(x_1,\dots,x_m)$ holds. This extends the proved two-factor case $RHR(x,y)$ to arbitrary products of Soergel bimodules; the paper presents it as the conjectural generalization of its main theorem.