Prime-folding characterization of extremal tensors

Let $2\le n=\prod_{k=1}^{s_n}p_k$ be the prime factorization, with $2\le p_1\le p_2\le\dots\le p_{s_n}$. Consider a tensor $\mathcal{T}\in\mathbb{B}^{p_1\times p_2\times\dots\times p_{s_n}\times p_1\times p_2\times\dots\times p_{s_n}}$, where $\mathbb{B}$ denotes the set of zero-one tensors. Prime-folding conjecture. If $\mathcal{T}$ satisfies

$$\|\mathcal{T}\|_\sigma=1\quad\text{and}\quad\|\mathcal{T}\|=\sqrt{n},$$

then $\mathcal{T}$ can be unfolded to a permutation matrix in $\mathbb{B}^{n\times n}$. This is the remaining structural step toward the preceding characterization conjecture, after reducing to maximum foldings with prime mode dimensions; the paper explicitly says that it cannot currently prove this assertion.