\textbf{Gaussian $\alpha$-projections are not coordinatewise variance-monotone} An exact counterexample to AIM problem 558, with a determinant-monotonicity theorem

For a fixed correlated Gaussian target, consider its optimal diagonal Gaussian approximation under $D_\alpha(p\Vert q)$, with $\alpha>1$. We give a rational three-dimensional target whose optimal covariance is the identity at $\alpha=10$, but whose second and third variances are strictly smaller than one for every $\alpha>10$. The first variance is strictly larger than one, so the covariances are incomparable in the coordinatewise order. A convex comparison argument proves the result without numerical optimization or limiting arguments. This disproves the statement in AIM problem 558 and the corresponding conjecture in Remark 12 of Margossian, Pillaud-Vivien and Saul. We also derive a general sensitivity formula, show that dimension three is minimal, and prove that the determinant of the optimal covariance nevertheless increases strictly for every non-diagonal Gaussian target and every $\alpha>1$.