Kährström's equivalence conjecture for Kostant-negative involutions

For $n\geq 0$, let $S_n$ be the symmetric group, let $\mathrm{Inv}_n$ be its set of involutions, and let $\mathbf{K}_n(w)=-$ mean that Kostant's problem for the simple highest weight module $L_w$ is negative. Let $\theta_x$ denote the projective functor indexed by $x\in S_n$, let $\mathcal{O}_0$ be the principal block of category $\mathcal{O}$, and let ${}^{\mathbb{Z}}\mathcal{O}_0$ be its graded version. Write $\underline{\hat{H}}_w$ and $\underline{H}_x$ for the relevant Kazhdan–Lusztig basis elements of the Hecke algebra, with specialization at $v=1$ denoted by $\left.\cdot\right|_{v=1}$. Kährström's conjecture. For $w\in\mathrm{Inv}_n$, the following conditions are equivalent: (I) $\mathbf{K}_n(w)=-$; (II) there exist different $x,y\in S_n$ such that $\theta_xL_w\cong\theta_yL_w\neq0$ in $\mathcal{O}_0$; (III) the same holds in ${}^{\mathbb{Z}}\mathcal{O}_0$; (IV) there exist different $x,y\in S_n$ such that $\left.\underline{\hat{H}}_w\underline{H}_x\right|_{v=1}=\left.\underline{\hat{H}}_w\underline{H}_y\right|_{v=1}\neq0$; and (V) $\underline{\hat{H}}_w\underline{H}_x=\underline{\hat{H}}_w\underline{H}_y\neq0$. The conjecture relates Kostant negativity to repeated nonzero images under projective functors and to corresponding Hecke-algebra identities; its status is not resolved by the supplied material.