On a determinantal inequality arising from diffusion tensor imaging

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$\det(A^2+|BA|)\le \det(A^2+AB),$$ where $A, B$ are $n\times n$ positive semidefinite matrices. We complement his result by proving $$\det(A^2+|AB|)\ge \det(A^2+AB).$$ Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.