Optimal mixed-norm inequality for unimodular multilinear forms

Let $m,n$ be positive integers, let $p\in[1, fty]$, and let $T:\ell_p^n\times\cdots\times\ell_p^n\to\mathbb{K}$ be a unimodular $m$-linear form. The norm $\|T\|$ is its operator norm. Optimal exponent conjecture. There is a constant $K_m$ such that

$$\left\vert \begin{array}{l} \displaystyle\left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{\frac{mp}{p-1}}\right) ^{\frac{p-1}{mp}}\leq K_{m}\left\Vert T\right\Vert \text{ for } 1 \leq p\leq2,\\ \displaystyle\left(\sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq 1.3m^{0.365}\left\Vert T\right\Vert \text{ for }p\geq2, \end{array} \right.$$

and the exponents are sharp. This would determine the optimal exponents in the Gale–Berlekamp permutation-switching problem in higher dimensions; the relevant intermediate vector-valued multilinear summability problems remain open.