Talagrand's Gaussian anti-concentration conjecture for diffusion

Let $n\geq 1$ and let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb R^n$. For $t>0$, let $U_t$ denote the Gaussian diffusion operator, and for a measurable non-negative function $f$ write $\|f\|_1=\int |f|\,d\gamma_n$. Talagrand's conjecture. For every $t>0$, there exists a function $\psi_t:[1,\infty)\to[1,\infty)$ with

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

such that for every measurable $f:\mathbb R^n\to\mathbb R_+$ and every $\alpha>1$,

$$\gamma_n\left(\{x:U_tf(x)>\alpha\|f\|_1\}\right)\leq\frac{1}{\alpha\psi_t(\alpha)}.$$

The conjecture asserts that Gaussian diffusion prevents the heat content of a non-negative function from concentrating near a single high temperature. It was resolved positively in the paper, with a bound of order $\sqrt{\log\alpha}/(\log\log\alpha)^4$; sharper dimension-dependent bounds were previously known in each fixed dimension.