Eskenazis–Ivanisvili's dimension-free heat-semigroup lower bound

Let $(X,\|\cdot\|_X)$ be a $K$-convex Banach space. For $n\in\mathbb{N}$, let $\operatorname{Deg}_{\le d}$ denote the space of functions $f:\{-1,1\}^n\to X$ whose Hamming cube Fourier–Walsh expansion has no terms of degree greater than $d$. Let $\Delta$ be the Hamming cube Laplacian and let $e^{-t\Delta}$ be its heat semigroup, defined by $e^{-t\Delta}w_S=e^{-t|S|}w_S$. Eskenazis–Ivanisvili's conjecture. For every $p\in(1,\infty)$, there exist constants $c(p,X)$ and $C(p,X)$ such that, for every $d\in[n]$, every $f\in\operatorname{Deg}_{\le d}$ and every $t\ge0$,

$$\|e^{-t\Delta}f\|_{L^p(\{-1,1\}^n;X)}\ge c(p,X)\,\exp(-C(p,X)\,td)\,\|f\|_{L^p(\{-1,1\}^n;X)}.$$

This is a dimension-free lower bound quantifying how much the heat semigroup can damp a low-degree function. The source says that it gives partial answers in the special case where $f$ takes values in $\{-1,0,1\}$ and $X=\mathbb{C}$, so the general statement remains open in the paper's presentation.