Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{4}),$$ seems to be only the start of a bunch of similar ones, as already suggested in a comment of the OP.
Experimentally, the following generalization seems to hold, of which the above is the $k=1$ case : Putting $C :=ABBA$ and $D :=BAAB$, we have for $k\in\mathbb N$,
$$\det (A^{4k}+ C^k+ D^k+B^{4k})\ge\det(A^{2k}+B^{2k})^2.$$ Further, still experimentally, $$\det (A^{16}+ CDDC+DCCD+B^{16})\ge\det(A^{8}+B^{8})^2,$$ which sort of iterates the way the multiplications are done (think of the Thue-Morse sequence), and, now combining both methods,$$\det (A^{16k}+ C^kD^kD^kC^k+D^kC^kC^kD^k+B^{16k})\ge\det(A^{8k}+B^{8k})^2,$$ which may certainly be further iterated.
On the other hand, all my attempts of generalizing in a straightforward way to inequalities like $\det(A^6+ABBAAB+BAABBA+B^6)\ge \det(A^{3}+B^{3})^2$ have failed. In fact, there seem to be only very few permutations of the $A$’s and $B$’s in the two middle terms of the LHS that yield valid inequalities, and it looks like in those permutations the $A$’s and $B$’s must be highly ’balanced’ in a specific way. I haven’t been lucky either with finding inequalities of similar types for more than 2 matrices.
Supposedly, Terry Tao’s sophisticated proof of the original inequality using eigenspaces and traces may be adapted to the above generalizations, though it might become very technical. But maybe:
Are there any insights why the feasible patterns of $A$’s and $B$’s on the LHS are what they are?
