Diagonal matrices maximize log-minor variance

Let $\mathcal M_{\kappa}$ be the set of positive-definite $n\times n$ matrices with condition number $\kappa$. For $k<n$, $\kappa$, and $M\in\mathcal M_{\kappa}$, define $Y_k(M)$ as the log-minor random variable associated with $M$. Diagonal matrices maximize log-minor variance. For all $k<n$, $\kappa$, and $M\in\mathcal M_{\kappa}$, there exists a diagonal matrix $D\in\mathcal M_{\kappa}$ such that

$$\operatorname{var}(Y_k(M))\leq\operatorname{var}(Y_k(D)).$$

The conjecture is motivated by the sharpness of the variance bound for diagonal matrices and examples suggesting that diagonal matrices attain the largest possible log-minor variance among positive-definite matrices with fixed condition number. Its resolution would identify the extremizers for the general variance bound.