I wonder if the following inequality involving skew symmetric matrices is true:

Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ is positive-definite. Then,

$$\mbox{Tr}(B^2 \Sigma C^2) - \frac{1}{2}\mbox{Tr}((CB) \Sigma (CB) + (CB) \Sigma (BC)) \geq 0 $$

For $\Sigma = I_d$, this is a relatively well-known inequality due to Bellman (and also follows from Araki-Lieb-Thirring). Not sure what can be said for $\Sigma$ beyond identity.

Any relevant tools/inequalities appreciated!