Gaussian comparison conjecture for maxima under variance and increment domination

Let $\mathbf{X}=(X_1,\ldots,X_d)$ and $\mathbf{Y}=(Y_1,\ldots,Y_d)$ be $d$-dimensional Gaussian vectors with common mean $\bm{\mu}=(\mu_1,\ldots,\mu_d)$ and covariance matrices $\Sigma^X$ and $\Sigma^Y$, whose $ij$ entries are $\sigma_{ij}^X$ and $\sigma_{ij}^Y$, respectively. Define

$$\gamma_{ij}^X=\mathbb{E}\{(X_i-X_j)^2\},\qquad \gamma_{ij}^Y=\mathbb{E}\{(Y_i-Y_j)^2\}.$$

Let $\operatorname{med}(\max_iY_i)$ denote the median of $\max_{1\leq i\leq d}Y_i$, namely the value $a$ satisfying $\Pr(\max_{1\leq i\leq d}Y_i\leq a)=0.5$. Suppose that $\sigma_{ii}^Y\geq\sigma_{ii}^X$ for every $i$ and $\gamma_{ij}^Y\geq\gamma_{ij}^X$ for every $i,j$. For any $c\geq\operatorname{med}(\max_iY_i)$, Gaussian comparison conjecture. There is a comparison bound of the form

$$\Pr\left(\max_{1\leq i\leq d}X_i\geq c\right)\leq (?)\;\Pr\left(\max_{1\leq i\leq d}Y_i\geq c\right).$$

Under these assumptions, the Sudakov--Fernique inequality gives only the expectation comparison $\mathbb{E}\{\max_iX_i\}\leq\mathbb{E}\{\max_iY_i\}$; the conjectured tail comparison would yield the desired asymptotic sharp-dominance result for the multivariate one-sided Gaussian-prepivoted test. It is stated to be true in the univariate case, while the multivariate case remains open.