Monotonicity conjecture for Gaussian noise stability at fixed volume

For $a\in(0,1)$, let $J(a,a;\rho)$ denote the Gaussian noise-stability quantity for two sets of Gaussian measure $a$, and let $\mathbb{E}_\rho J(1_A(X),1_A(Y);\rho)$ denote the corresponding quantity for a set $A\subset\mathbb{R}^n$, where $X$ and $Y$ are standard Gaussian vectors with correlation $\rho$. The conjecture asserts that there is a function $k(\rho,a)$ such that, for every fixed $a\in(0,1)$,

$$k(\rho,a)\sim\sqrt{1-\rho}\quad\text{as }\rho\to1,$$

and

$$k(\rho,a)\sim\rho\quad\text{as }\rho\to0,$$

while, for every $a\in(0,1)$ and every $A\subset\mathbb{R}^n$,

$$\frac{J(a,a;\rho)-\mathbb{E}_\rho J(1_A(X),1_A(Y);\rho)}{k(\rho,a)}$$

is increasing in $\rho$. This would remove the half-measure and integrality restrictions in the proposition used to compare Gaussian noise parameters, potentially improving the quantitative robustness results and extending them to arbitrary fixed volumes.