Consider the following block matrix $$ X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix}, $$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}$.

**To Prove (or disprove)**: If $X$ is positive definite, i.e. $X>0$, then the following trace inequality holds
$$
\left[\mathrm{tr}(CC^\top)\right]^2< \mathrm{tr}(A^2)\mathrm{tr}(B^2).
$$

**Some comments.** Based on Theorem 2.3 of Horn and Mathias. "Cauchy-Schwarz inequalities associated with positive semidefinite matrices." *Linear Algebra and Its Applications* 142 (1990): 63-82, I think it is possible to prove the previous fact if we replace strict inequalities with non-strict ones.