Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$

\begin{align*} C = A^TXA + B^TXB. \end{align*}

In my case, the coefficient matrices are such that the equation has a unique real solution which is also symmetric.

I have been trying to prove that the solution must also be positive definite and I am struggling. The struggle makes me think that this is actually not true.

I would be grateful for either a pointer to related literature, a way to prove my hypothesis or a counterexample. If it is not true in general, are there any known conditions on $A$ and $B$ which make this true?