The p-distinguished involution conjecture

Let $W$ be a Weyl group with $p$-canonical basis $\{c_w\}$ and left $p$-cells. For $w,x,y\in W$, let $\mu_{w,x}^y$ be the coefficient of $c_y$ in $c_wc_x$, and let $h_x^y$ be the coefficient of $H_y$ in the standard-basis expansion of $c_x$. For a Laurent polynomial $f$, write $\operatorname{val}(f)$ for its smallest exponent with nonzero coefficient.

The p-distinguished involution conjecture. In each left $p$-cell there is a unique involution $d$ such that

$$-\operatorname{val}(\mu_{d,d}^d)\geq \operatorname{val}(h_d^1).$$

This is presented as a conjectural analogue of distinguished involutions for $p$-cells. The paper notes that the theory of $p$-distinguished involutions is not fully developed.