Kährström's conjecture for involutions in category O

Let $\mathrm{Inv}_n$ be the set of involutions in $\mathrm S_n$, let $L_z^{(n)}$ be the simple highest weight module indexed by $z$, let $\theta_x^{(n)}$ be the projective functor indexed by $x$, and let $D_z^{(n)}C_x^{(n)}$ denote the corresponding Hecke-algebra product. Write $\leq_R^{(n)}$ for the right Kazhdan–Lusztig preorder, and use the graded category {}^^{\mathbb Z}\mathcal O_0^{(n)}, its ungraded version $\mathcal O_0^{(n)}$, and the Hecke algebras $\mathrm H_n^{\mathbb A}$ and $\mathrm H_n^{\mathbb Z}$. Kährström's Conjecture. For any involution $z\in\mathrm{Inv}_n$, the following are equivalent: (1) $z$ is Kostant positive; (2) for distinct $x,y\leq_R^{(n)}z^{-1}$, $\theta_x^{(n)}L_z^{(n)} ot\cong\theta_y^{(n)}L_z^{(n)}$ in the graded category; (3) the same nonisomorphism holds in the ungraded category; (4) $D_z^{(n)}C_x^{(n)} eq D_z^{(n)}C_y^{(n)}$ in $\mathrm H_n^{\mathbb A}$; and (5) the same inequality holds in $\mathrm H_n^{\mathbb Z}$. This conjecture relates Kostant positivity to distinctness of projective-functor images and their decategorified Hecke-algebra classes. Its status is not resolved in the supplied text; the paper presents it as a conjecture and proves related results in specified cases.