Optimal dimension exponents for coefficient inequalities of unimodular multilinear forms

Let $m$ be a positive integer, let $(r,p)\in(0,\infty)\times(1,\infty]$, and let $T:\ell_p^n\times\cdots\times\ell_p^n\to\mathbb{K}$ be a unimodular $m$-linear form, where $n$ is a positive integer. The norm $\|T\|$ is its operator norm. Optimal dimension-exponent conjecture. There is a constant $K_m$ such that

$$\left\vert \begin{array}{l} \displaystyle\left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{r}\right) ^{\frac{1}{r}}\leq K_{m}n^{\max\{\frac{mp+r-pr}{pr},0\}}\left\Vert T\right\Vert\text{ for }1< p\leq2, \\ \displaystyle\left(\sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{r}\right) ^{\frac{1}{r}}\leq 1.3m^{0.365}n^{\max\{\frac{2mr+2mp-mpr-pr}{2pr},0\}}\left\Vert T\right\Vert\text{ for }p\geq2, \end{array} \right.$$

and the exponents $\max\{\frac{2mr+2mp-mpr-pr}{2pr},0\}$ and $\max\{\frac{mp+r-pr}{pr},0\}$ are sharp. This is presented as a consequence that would follow if the preceding conjecture were correct, so its status is conditional and open.