Talagrand's convolution conjecture for the Boolean hypercube

Consider the Boolean hypercube $\left\{-1,1\right\}^n$ with its uniform probability measure $\mu$. For $t\geq 0$, let $\xi_t$ have independent coordinates satisfying

$$\mathbb{P}(\xi_t^{[i]}=1)=\frac{1+e^{-t}}{2},\qquad \mathbb{P}(\xi_t^{[i]}=-1)=\frac{1-e^{-t}}{2},$$

and define the heat semigroup by

$$P_t f(x):=\mathbb{E}[f(x\odot\xi_t)].$$

For a function $f:\left\{-1,1\right\}^n\to\mathbb{R}$, write $\|f\|_p=\left(\int |f|^p\,d\mu\right)^{1/p}$. Talagrand's convolution conjecture. For every $\tau>0$, there exists a constant $c_\tau>0$ depending only on $\tau$, such that for every nonnegative function $f:\left\{-1,1\right\}^n\to\mathbb{R}_+$ with $\|f\|_1\neq 0$, and every $\eta>1$,

$$\mathbb{P}_{X\sim\mu}\left(P_\tau f(X)>\eta\|f\|_1\right)\leq\frac{c_\tau}{\eta\sqrt{\log\eta}}.$$

The conjecture predicts a dimension-free gain of a factor $1/\sqrt{\log\eta}$ over Markov's inequality, quantifying the regularization produced by convolution on $L^1$ functions. The paper proves the bound up to a factor of $\log\log\eta$, while the stated conjecture remains unresolved.