Q-key expansion conjecture for orthogonal involution Schubert polynomials

Let $I_\infty$ denote the involutions of the infinite symmetric group, let $\mathfrak S^{\mathsf{O}}_z$ be the corresponding orthogonal involution Schubert polynomial, and let $\mathsf{cyc}(z)$ be the number of nontrivial cycles of $z$. Q-key expansion conjecture. For every $z\in I_\infty$, there is a finite set $\mathcal Y(z)$ of symmetric weak compositions such that

$$\mathfrak S^{\mathsf{O}}_z=\sum_{\alpha\in\mathcal Y(z)}2^{\mathsf{cyc}(z)-\mathsf{diag}(\alpha)}\kappa^\mathsf{Q}_\alpha,$$

where the Q-key polynomials are distinct and the coefficients are integral powers of two. The conjecture was verified computationally for $z\in I_n$ with $n\leq9$, but remains unresolved in general.