Transpose symmetry conjecture for generalized Kostka polynomials

Let $R$ be a dominant sequence of rectangles, and let $R^t$ be obtained by transposing every rectangle in $R$. Let $R'$ be a dominant rearrangement of $R^t$, and let $m$ be the total number of columns in the rectangles of $R$. Transpose symmetry conjecture.

$$K_{\lambda^t;R'}(q)=\widetilde{K}_{\lambda;R}(q),$$

where the left-hand side is computed in $GL(m)$. This conjecture relates transposition of shapes and rectangles to the cocharge normalization; the source calls the property mysterious and notes that it is not evident from the defining modules.