DeWitt's generalized identity for shifted stable Grothendieck polynomials

Let $m,k\in\mathbb{Z}_{>0}$, let $\mu=(m^k)$, and let $\nu=(m+k-1,m+k-3,\dots,|m-k|+1)$. For partitions $\lambda$ and $\mu$ with $\mu\subseteq\lambda$, let $G\hspace{-0.2mm}S^{(\beta)}_{\lambda/\mu}$ denote the $K$-theoretic Schur $S$-function, and let $G\hspace{-0.2mm}Q^{(\beta)}_\nu$ denote the shifted stable Grothendieck polynomial associated with the strict partition $\nu$. DeWitt's generalized identity. If $\mu$ and $\nu$ are as above, then

$$G\hspace{-0.2mm}S^{(\beta)}_{\mu}=G\hspace{-0.2mm}Q^{(\beta)}_{\nu}.$$

At $\beta=0$, this specializes to DeWitt's identity $S_\mu=Q_\nu$. The conjecture proposes that this identity persists for the corresponding $K$-theoretic shifted stable Grothendieck polynomials.