The inverse problem for generalized contraharmonic means

In this paper we introduce the generalized contraharmanic mean associated to a Kubo-Ando mean $\sigma$ as
$$ C_\sigma(X, Y) = X\sigma Y - X\sigma^\perp Y, $$
where $\sigma^\perp$ is the dual mean of $\sigma$ and $X, Y$ are positive definite matrices. We show that for a symmetric Kubo-Ando mean $\sigma$ such as $\sigma \ge \sharp$ and for any positive definite matrices $A \ge B$ the inverse problem
\begin{equation*} A=C_\sigma(X, Y), \ \ B=X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2} \end{equation*}
has a positive solution $(X, Y)$.