Staircase conjugation conjecture for Grothendieck polynomials

Let $\delta_n=\langle n-1,n-2,\ldots,1\rangle$ be the staircase partition, and let $\mu$ be a diagram contained in $\delta_n$. For a skew shape $A$, write $A^T$ for its transpose. Staircase conjugation conjecture.

$$g_{\delta_n/\mu}=g_{(\delta_n/\mu)^T}\qquad\text{and}\qquad G_{\delta_n/\mu}=G_{(\delta_n/\mu)^T}.$$

Computation strongly suggests these identities, which would extend the corresponding Schur equivalence; no proof or general resolution is supplied.