The generalized Cauchy–Bunyakovsky–Schwarz inequality conjecture for hypermatrices

Let $m,n\geq1$, let ${\bf d}=(d_1,\ldots,d_m)$, and let $\mathcal{M}_{\bf d}$ be the space of complex hypermatrices of type $d_1\times\cdots\times d_m$. For $x^{(k)},u^{(k)}\in\mathcal{M}_{\bf d}$, define

$$\Phi_{\bf d}^{(n)}(x^{(1)},\ldots,x^{(n)},u^{(1)},\ldots,u^{(n)})$$

by the generalized contraction formula given in the source. Cauchy–Bunyakovsky–Schwarz conjecture. The inequality

$$\Phi_{\bf d}^{(n)}(x^{(1)},\ldots,x^{(n)},u^{(1)},\ldots,u^{(n)})\geq0$$

is valid for all dimension vectors ${\bf d}$, all $m,n\geq1$, and all complex hypermatrices $x^{(k)},u^{(k)}\in\mathcal{M}_{\bf d}$.

For $n=2$, this includes the hypermatrix inequality equivalent to the Generalized Distillability Conjecture. Its validity for all numbers of factors, dimensions, and summands is presented as an open strengthening.