The vector-valued Borell inequality conjecture

Let $k$ be a positive integer and let $f\colon\mathbb{R}^{k}\to S^{k-1}$ be measurable, where $S^{k-1}=\{x\in\mathbb{R}^{k}:\|x\|=1\}$. Define $f_{\rm opt}(x)=x/\|x\|$ for $x\in\mathbb{R}^{k}\setminus\{0\}$. For $-1<\rho<1$, let $T_{\rho}$ be the Ornstein–Uhlenbeck operator and let the noise stability of $f$ be

$$\int_{\mathbb{R}^{k}}\langle f(x),T_{\rho}f(x)\rangle\gamma_{k}(x)\,\mathrm{d}x.$$

Vector-valued Borell inequality. If $0<\rho<1$ and

$$\int_{\mathbb{R}^{k}}f(x)\gamma_{k}(x)\,\mathrm{d}x=0,$$

then

$$\int_{\mathbb{R}^{k}}\langle f(x),T_{\rho}f(x)\rangle\gamma_{k}(x)\,\mathrm{d}x\leq\int_{\mathbb{R}^{k}}\langle f_{\rm opt}(x),T_{\rho}f_{\rm opt}(x)\rangle\gamma_{k}(x)\,\mathrm{d}x.$$

If $-1<\rho<0$, the inequality is reversed. In either case, equality holds only if there is an orthogonal $k\times k$ matrix $M$ such that $f(x)=f_{\rm opt}(Mx)$ for almost every $x\in\mathbb{R}^{k}$. This is a vector-valued extension of Borell's noise-stability inequality; the positive-correlation case includes the zero-mean condition, while the negative-correlation case does not. The source presents the assertion as a conjecture, and its resolution is not established in the supplied material.