Uniqueness of the unitary group representation minimizer for HyperCube regularization

Let $D$ be a group operation table, and let $T$ be the tensor represented by the HyperCube factors. For a finite group, write $\varrho$ for its unitary regular representation, so that the factor slices satisfy

$$A_g=B_g=C_g^\intercal=\varrho(g).$$

Let $\mathcal{H}$ denote the HyperCube Regularizer and impose the constraint $T=D$.

HyperCube regularizer minimizer conjecture. The unitary group representation $A_g=B_g=C_g^\intercal=\varrho(g)$ describes the unique minimizer of $\mathcal{H}$ up to unitary basis changes, and its minimum regularizer loss is

$$\mathcal{H}^*(D)=3\lVert D\rVert_F^2.$$

This conjecture proposes that the regular representation is not only sufficient to reproduce every finite group operation table exactly, but is also the unique optimal solution, modulo unitary changes of basis, for the constrained HyperCube regularization problem.