Monotonicity of the p-cell a-function under the two-sided p-preorder

Let $W$ be the Weyl group under consideration, and let ${}^{p} \!\mkern 1mu {\mathbf a}_{\mathbf c}:W\to\mathbb{R}_{\ge 0}$ be the function obtained by extending the cell invariant ${}^{p} \!\mkern 1mu {\mathbf a}_{\mathbf c}$ from left $p$-cells to their elements. Write $x\mathrel{\overset{p}{\underset{2}{\leqslant}}y}$ for the two-sided $p$-preorder on $W$. Monotonicity conjecture. For $x,y\in W$,

$$x\mathrel{\overset{p}{\underset{2}{\leqslant}}y}\quad\Rightarrow\quad{}^{p} \!\mkern 1mu {\mathbf a}_{\mathbf c}(x)\geqslant{}^{p} \!\mkern 1mu {\mathbf a}_{\mathbf c}(y).$$

The preceding corollary establishes that this function is constant on two-sided $p$-cells; the conjecture asks for its monotonicity along the two-sided $p$-preorder, an analogue of the corresponding monotonicity phenomenon for Kazhdan–Lusztig cell invariants.