Fourier-matrix decomposition conjecture for entropy diagrams

Let $d=d_1d_2$ be composite, and let $U_d^F$ denote the Fourier matrix linking the two observables. Consider also the tensor product $U_{d_1}^F\otimes U_{d_2}^F$.

Fourier-matrix decomposition conjecture. Replacing $U_d^F$ by $U_{d_1}^F\otimes U_{d_2}^F$ does not change the entropy diagram.

Although the Fourier matrix need not decompose as a tensor product of smaller Fourier matrices, this conjecture would imply that the curve of minimal entropy pairs is unchanged by the replacement and would simplify the composite-dimensional problem.