I'd like to have the equality condition in the Araki–Lieb–Thirring inequality $$\operatorname{Tr} [(BAB)^r]\leq \operatorname{Tr} [(B^{r}A^{r}B^{r})],$$ valid for $A,B$ semidefinite positive and $r\geq1$ (I'm interested in the case $r=2$). Any clue? I didn't find the proof of it yet, I guess it would be a good start!

As explained in comment, finding the equality condition for the inequality $$\operatorname{Tr} [X^2]\leq \operatorname{Tr} [XX^\dagger]$$ is sufficient!

It is the inequality n°13 of Lieb and Thirring - Inequalities for the moments of the Eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities.