Characterization of extremal nonnegative tensors by unfolded permutation tensors

Let $\boldsymbol{\beta}$ denote the lower bound in equation (main) for the ratio of the spectral norm to the Frobenius norm. An unfolded permutation tensor is a tensor obtained by unfolding a permutation matrix, as defined in the surrounding discussion; multiplication by a positive constant means replacing $\mathcal{T}$ by $c\mathcal{T}$ for some $c>0$. Characterization conjecture. The lower bound $\boldsymbol{\beta}$ is achieved by a nonnegative tensor if and only if that tensor is an unfolded permutation tensor up to multiplication by a positive constant. This would give a complete characterization of the tensors attaining the extreme spectral-to-Frobenius norm ratio; the paper states that this remains unresolved and develops a reduction to the case of maximum foldings with prime mode dimensions.