General correspondence between real and complex solutions of the matrix equation

Let equation be the complex matrix equation considered in the paper, and let equation be its associated real matrix equation. If $\boldsymbol{Y}$ is the general real solution of equation, let $\boldsymbol{X}^{\prime}$ be the matrix defined in equation. Solution-correspondence conjecture. If $\boldsymbol{Y}$ is the general real solution of equation, then the general solution of equation is $\boldsymbol{X}^{\prime}$. The authors state that they cannot prove this assertion in general, so its validity requires further study.