Lin's symmetry-preserving almost-commuting matrices conjecture

Let $H$ and $X$ be self-adjoint matrices in a matrix algebra, and suppose that $(X,H)$ belongs to an Altland–Zirnbauer symmetry class. Lin's theorem states that sufficiently almost-commuting $H$ and $X$ can be approximated by commuting self-adjoint matrices $H_1$ and $X_1$. Lin's symmetry-preserving conjecture. Lin's theorem should remain true when $(X,H)$ belongs to any Altland–Zirnbauer symmetry class, with the approximating pair $(X_1,H_1)$ required to belong to the same symmetry class. The claim asks whether Lin's approximation can preserve the imposed time-reversal, particle-hole, or chiral symmetry; the supplied text gives no resolution.