Conjugation automorphism conjecture for the G-positivity order

For skew shapes $A$ and $B$, let $A\leq B$ mean that $a_{A,\nu}\leq a_{B,\nu}$ for every partition $\nu$, equivalently that $G_B-G_A$ is $G$ positive. Transposition sends a skew shape $A$ to its transpose $A^T$. Conjugation automorphism conjecture. Conjugation acts as an automorphism of this $G$-positivity order:

$$A\leq B\quad\Longleftrightarrow\quad A^T\leq B^T.$$

The claim is proposed as an analogue of the corresponding fact for the Schur-positivity order; the source gives no proof or resolution.