Fourier completely bounded polynomial influential-variable conjecture

Let $p:\{-1,1\}^n\to\mathbb{R}$ be a polynomial of degree at most $d$. Its Fourier completely bounded $d$-norm is denoted by $\|p\|_{\operatorname{fcb},d}$, and $\operatorname{Var}[p]$ denotes its variance. Fourier completely bounded influential-variable conjecture. If $\|p\|_{\operatorname{fcb},d}\leq 1$, then $p$ has a variable with influence at least $\operatorname{poly}(\operatorname{Var}[p],1/d)$. This is proposed as a weaker conjecture than the Aaronson–Ambainis conjecture and would also imply the folklore conjecture concerning classical simulation of quantum query algorithms on most inputs. Its status is not specified in the source.