The asymmetric Cauchy–Schwarz complexity conjecture

Let $\Phi=(\phi_1,\dots,\phi_k)$ be a system of linear forms, with $\phi_i\colon\mathbb{F}_p^d\to\mathbb{F}_p$, and let $i_0\in[k]$. Let $s(\Phi,i_0)$ be the asymmetric true complexity of the distinguished form. Asymmetric Cauchy–Schwarz conjecture. There should exist $M\geq0$ such that, for every $n\geq1$ and every collection of $1$-bounded functions $f_1,\dots,f_k\colon\mathbb{F}_p^n\to\mathbb{C}$, one has

$$\left\lvert\Lambda_{\Phi}(f_1,\dots,f_k)\right\rvert\leq\|f_{i_0}\|_{U^{s(\Phi,i_0)+1}}^{2^{-M}}.$$

The claim also requires that this bound be provable using only repeated applications of Cauchy–Schwarz. The symmetric analogue is handled by the paper's main theorem, whereas natural systems satisfying the asymmetric statement remain open; the conjectured proof method is intended as a test case for strengthening Cauchy–Schwarz arguments.