Optimal asymptotic estimate for unimodular multilinear forms on sequence spaces

Let $d$ be the multilinearity degree, let $p_{1},\dots,p_{d}\in[1,\infty]$, and let $A:\ell_{p_{1}}^{n_1}\times\cdots\times\ell_{p_{d}}^{n_d}\to\mathbb{K}$ be a unimodular multilinear form, meaning that all its coefficients have the same modulus. Define

$$\gamma:=\min\left\{2,\max\{p_k:p_k\leq2\}\right\}.$$

Optimal asymptotic estimate. There exist constants $B_d,C_d<\infty$ such that

$$B_d\leq\inf\frac{\lVert A\rVert}{\left(\sum_{k=1}^{d}n_k^{1-\frac{1}{\gamma}}\right)\cdot\prod_{k=1}^{d}n_k^{\max\left(\frac{1}{\gamma}-\frac{1}{p_k},0\right)}}\leq C_d,$$

for all such unimodular multilinear forms $A$, and the exponents involved are sharp. The infimum is taken over the relevant unimodular forms.