Shifted $J$-Grothendieck expansion conjecture

Let $\lambda$ be a strict partition. Let $\widehat{\mathbf{JP}}{}^+$ be the set of infinite $\mathbb{Z}_{\geq 0}[\beta]$-linear combinations of the functions $JP^{(\beta)}_\mu$, and let $\mathbf{gq}^+$ be the indicated positive span of the functions $gq^{(\beta)}_\mu$.

Shifted $J$-Grothendieck expansion conjecture. If $\lambda$ is a strict partition, then

$$GP^{(\beta)}_\lambda\in \widehat{\mathbf{JP}}{}^+$$

and

$$jq^{(\beta)}_\lambda \in \mathbf{gq}^+.$$

This is presented as a shifted analogue of known positive expansions between stable Grothendieck functions. The supplied status marks the conjecture as open.