Let $X$ and $L$ be real positive definite matrices.
$$\operatorname{Trace}(X^{-1}(X - e^{\log(X) - L})^2) \leq \operatorname{Trace}(XL^2)$$
where the exponential and the log are matrix exponential and log (defined as applying those functions to the eigenvalues)
I think I have for sanity check checked basic cases like when these are 1x1 or which basically implies the inequality for the case when X and L are diagonal which in turn implies it for the case when X and L commute.
I would love to have a simple proof if it is true. Even if this is true under some mild conditions on L (like spectral norm $\le$ constant) that is fine too.
Thanks