The perturbed-metric conjecture for Gaussian noise stability deficit

Let $0<\rho<1$ and let $A\subset\mathbb R^n$ be measurable with $0<\gamma^n(A)<1$. Write $v=v(H(A))/|v(H(A))|$, and let $\mu$ and $\nu$ be the push-forwards under $x\mapsto\langle v,x\rangle$ of Gaussian measure restricted to $A$ and $H(A)$, respectively. If $f$ and $g$ are their densities with respect to $\gamma^1$, define

$$\tilde \varepsilon_\rho(A)=\left|\int_{\mathbb R}\Phi\left(\frac{\rho x-\alpha}{\sqrt{1-\rho^2}}\right)(g(x)-f(x))\,d\gamma^1(x)\right|.$$

The perturbed-metric conjecture. For every $0<s<1$, there exist constants $C_s,c_s>0$ such that, whenever $\varepsilon(A)<e^{-1/\rho}$,

$$C_{\gamma(A)}\tilde \varepsilon_\rho(A)\geq \mathcal{S}_\rho(H(A))-\mathcal{S}_\rho(A)\geq c_{\gamma(A)}\tilde \varepsilon_\rho(A)(1-\rho).$$

In particular, the Gaussian noise stability deficit should be equivalent, up to constants depending only on $\rho$ and $\gamma(A)$, to an expression depending only on the marginal of $A$ in the direction $v$. The conjecture is motivated by the possibility that a slightly perturbed metric removes the logarithmic loss caused by tail phenomena; its resolution is not supplied in the source.