Equality of nonnegative and zero-one tensor spectral-to-Frobenius ratios

For a tensor dimension $n_1\times n_2\times\dots\times n_d$, let $\phi(\mathbb{R}^{n_1\times n_2\times\dots\times n_d}_+)$ be the infimum of the spectral-to-Frobenius norm ratio over nonnegative real tensors, and let $\phi(\mathbb{B}^{n_1\times n_2\times\dots\times n_d})$ be the corresponding quantity over zero-one tensors. Ratio equality conjecture.

$$\phi(\mathbb{R}^{n_1\times n_2\times\dots\times n_d}_+)=\phi(\mathbb{B}^{n_1\times n_2\times\dots\times n_d}).$$

The authors describe this as a bold conjecture motivated by exhaustive searches for small dimensions and believe the resulting upper bounds are tight; exact values for the other small dimensions were not established in the paper.