Dimension-free quantitative Gaussian isoperimetric inequality

Let $E\subset\mathbb{R}^n$ be an open set with Lipschitz boundary and Gaussian measure $\gamma(E)=\phi(s)$, where

$$\phi(s):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^s e^{-t^2/2}\,dt.$$

Its Gaussian perimeter is

$$P_\gamma(E):=\frac{1}{(2\pi)^{(n-1)/2}}\int_{\partial E}e^{-|x|^2/2}\,d\mathcal{H}^{n-1}(x),$$

and define the Gaussian isoperimetric deficit and Fraenkel asymmetry by

$$D(E):=P_\gamma(E)-e^{-s^2/2},\qquad \alpha(E):=\min_{\omega\in\mathbb{S}^{n-1}}\gamma(E\triangle H_{\omega,s}),$$

where $H_{\omega,s}:=\{x\in\mathbb{R}^n:x\cdot\omega<s\}$ and $\triangle$ denotes symmetric difference. Dimension-free quantitative Gaussian isoperimetric conjecture. The inequality

$$\alpha(E)^2\leq c(s)D(E)$$

holds for a constant $c(s)$ depending only on the mass parameter $s$, and not on the dimension $n$. This conjecture asks for a dimension-independent strengthening of quantitative Gaussian isoperimetry; the corresponding estimate is known with a constant depending on $n$ and $s$, while the cited dimension-free result for $s=0$ has a sub-optimal exponent.