Crosshatch-pair conjecture for equivalence classes of Kazhdan–Lusztig pairs

Given a composition $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)\models n$, define the permutation

$$x_\alpha=[n-\alpha_1,n-\alpha_1+1,\ldots,n-1,n-\alpha_1-\alpha_2,n-\alpha_1-\alpha_2+1,\ldots,n-\alpha_1-1,\ldots,0,1,\ldots,\alpha_k-1].$$

Let $X_n=\{x_\alpha:\alpha\models n\}$. A crosshatch pair is a pair $x\leq w$ such that $xw_0,w\in X_n$. The relation $\sim$ is the equivalence relation on pairs induced by the L-S operators. Crosshatch-pair conjecture. Every $\sim$-equivalence class contains a crosshatch pair. The conjecture concerns representatives of equivalence classes and would give a structured representative for each class; the surrounding computations suggest further coalescence as the parameter $k$ increases, but no resolution of this conjecture is given.