Involution Schubert polynomial expansion conjecture in P- and Q-key polynomials

Let $z$ index an involution Schubert polynomial, with $\frac{}{}\mathfrak S^{\mathsf{Sp}}_z$ denoting the symplectic polynomial and $\mathfrak S^{\mathsf{O}}_z$ the orthogonal polynomial. Let $P$-key polynomials and $Q$-key polynomials be the shifted key-polynomial families introduced in the paper.

Involution Schubert expansion conjecture. Each involution Schubert polynomial $\mathfrak S^{\mathsf{Sp}}_z$ (respectively, $\mathfrak S^{\mathsf{O}}_z$) is an $\mathbb{N}$-linear combination of $P$-key polynomials (respectively, $Q$-key polynomials).

These expansions would extend the positivity phenomena known for stable involution Schubert polynomials and provide a finite shifted-key analogue of Schur $P$- and $Q$-function expansions. The source presents this as a conjecture and gives no resolution.