Talagrand's discrete-cube anti-concentration conjecture

Let $n\geq 1$ and $t\geq 0$, let $\mu_t=\frac{1-e^{-t}}{2}\delta_{-1}+\frac{1+e^{-t}}{2}\delta_1$, and let $\mu=\mu_\infty$ be the uniform probability measure on $\{-1,1\}$. Write $\mu^n$ for the product measure and let $T_t f=f*\mu_t^n$ on functions $f:\{-1,1\}^n\to\mathbb R$. For non-negative $f$, write $\|f\|_1$ for its $L^1(\mu^n)$ norm. Talagrand's conjecture. For every $t>0$, there exists a function $\varphi_t:[1,\infty)\to[1,\infty)$ with

$$\lim_{\alpha\to\infty}\varphi_t(\alpha)=\infty$$

such that for every $f:\{-1,1\}^n\to\mathbb R_+$ and every $\alpha>1$,

$$\mu^n\left(\{x\in\{-1,1\}^n:T_tf(x)>\alpha\|f\|_1\}\right)\leq\frac{1}{\alpha\varphi_t(\alpha)}.$$

This is presented as a discrete-cube generalization of the Gaussian conjecture and is proved by approximately embedding Gaussian space into discrete cubes of growing dimension via the central limit theorem.