Real equivariant almost-commuting sphere matrices conjecture

Let $X_1$, $Y_1$ and $H_1$ be matrices in $\boldsymbol{M}_{2n}(\mathbb{C})$ satisfying

$$X_1^{\dagger}=X_1=-\overline{X_1},\qquad Y_1^{\dagger}=Y_1=-\overline{Y_1},\qquad H_1^{\dagger}=H_1=\overline{H_1},$$

and

$$\mathopen{}\mathclose\bgroup\originalleft\Vert [H_1,X_1]\aftergroup\egroup\originalright\Vert\leq\delta,\qquad \mathopen{}\mathclose\bgroup\originalleft\Vert [H_1,Y_1]\aftergroup\egroup\originalright\Vert\leq\delta,\qquad \mathopen{}\mathclose\bgroup\originalleft\Vert X_1^2+Y_1^2+H_1^2-I\aftergroup\egroup\originalright\Vert\leq\delta.$$

Real equivariant almost-commuting sphere matrices conjecture. For every $\eta>0$ there is a $\delta>0$, independent of $n$, such that there is a triple $(X_2,Y_2,H_2)$ of commuting Hermitian matrices satisfying

$$X_2^{\rho}=-X_2,\qquad Y_2^{\rho}=-Y_2,\qquad H_2^{\rho}=H_2,$$

and

$$X_2^2+Y_2^2+H_2^2=I,$$

and lying within $\eta$ of the original triple:

$$\mathopen{}\mathclose\bgroup\originalleft\Vert X_2-X_1\aftergroup\egroup\originalright\Vert\leq\eta,\qquad \mathopen{}\mathclose\bgroup\originalleft\Vert Y_2-Y_1\aftergroup\egroup\originalright\Vert\leq\eta,\qquad \mathopen{}\mathclose\bgroup\originalleft\Vert H_2-H_1\aftergroup\egroup\originalright\Vert\leq\eta.$$

This asserts stable approximation of the real-symmetry-constrained approximate sphere relations by exactly commuting matrices. It is motivated by the analogous result for almost commuting real orthogonal matrices and would yield the corresponding classification result for the rotation-equivariant two-sphere setting if true.