Stable orthogonal Grothendieck specialization for involutions fixing 1

Let $I_\infty$ be the involutions of the infinite symmetric group, let $\mathrm{Des}_V(z)$ be the visible descent set, and let $\mathfrak G^{\mathsf{O}}_z$ be the orthogonal involution Grothendieck polynomial. Stable specialization conjecture. If $z\in I_\infty$ satisfies $z(1)=1$, then the limit in the source converges as a power series to a symmetric function and

$$GQ_z(x_1,x_2,\ldots,x_n)=\pi^{(\beta)}_{w_n}(\mathfrak G^{\mathsf{O}}_z)$$

for every $n\in\mathbb P$ such that $\mathrm{Des}_V(z)\subseteq[n]$. The supplied status is open, but the stated result is described as resolving an earlier question about finite positive expansions of $GQ_\mu$ into $G_\lambda$'s; the exact resolution status should be checked against the paper.