Shifted Grothendieck positivity conjecture for skew products

Let $\beta$ be the parameter, and let $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ be strict partitions. Let $GQ^{(\beta)}_{\lambda/\!/\mu}$ denote the corresponding skew shifted Grothendieck function, and let $\mathbf{GQ}^+$ and $\mathbf{gp}^+$ be the indicated positive spans of shifted Grothendieck functions.

Shifted Grothendieck positivity conjecture. One has

$$GQ^{(\beta)}_{\lambda/\!/\mu} \in \mathbf{GQ}^+$$

and

$$gp^{(\beta)}_{\lambda}gp^{(\beta)}_{\mu} \in \mathbf{gp}^+.$$

These are conjectural positivity properties analogous to the known positivity of products of the corresponding $GP^{(\beta)}$ and $GQ^{(\beta)}$ functions.