Albuquerque–Rezende's norm estimate conjecture for unimodular multilinear forms

Let $m\geq 1$ and let $p_{1},\dots,p_{m}\in[1,\infty]$. For positive integers $n_{1},\dots,n_{m}$, let $A:\ell_{p_{1}}^{n_{1}}\times\cdots\times\ell_{p_{m}}^{n_{m}}\rightarrow\mathbb{K}$ be a unimodular multilinear form, meaning that all coefficients of $A$ have modulus one. Define

$$\gamma:=\min\left\{2,\max\{p_{k}:p_{k}\leq 2\}\right\}.$$

The infimum below is taken over all such forms $A$.

Albuquerque–Rezende's conjecture. There exist constants $B_{m},C_{m}>0$, depending only on $m$, such that

$$B_{m}\leq\inf\frac{\lVert A\rVert}{\left(\sum_{k=1}^{m}n_{k}^{1-\frac{1}{\gamma}}\right)\cdot\prod_{k=1}^{m}n_{k}^{\max\left\{\frac{1}{\gamma}-\frac{1}{p_{k}},0\right\}}}\leq C_{m},$$

and the exponents involved are sharp.

The conjecture proposed an optimal two-sided estimate for the norm of unimodular multilinear forms on products of finite-dimensional sequence spaces. It is false: the source notes that a subsequent estimate disproves the conjecture.