Uniqueness correspondence for complex and real matrix equations

Let equation be the complex matrix equation and equation its associated real matrix equation. Denote the real representation of a solution $X$ by $X_{\sigma}$. Uniqueness-correspondence conjecture. If equation has a unique solution $X^{\star}$, then equation has a unique solution $Y^{\star}$. Moreover,

$$Y^{\star}=X_{\sigma}^{\star}.$$

The assertion is presented as a conjectural converse to Item 3 of Theorem and as a generalization of Theorem; the paper states that it cannot prove the claim in general.