Majorization conjecture for products of positive semidefinite matrices

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. Let $\lambda(X)$ denote the vector of eigenvalues of a Hermitian matrix $X$, arranged in decreasing order, and let $\succ$ denote majorization. Majorization conjecture.

$$\lambda(A^2+|BA|^p)\succ\lambda(A^2+|AB|^p)\quad\text{for all }p>0.$$

If true, this would imply the corresponding determinant inequality for all positive $p$ and would extend the known cases discussed immediately beforehand. The source reports numerical evidence but leaves the majorization statement as a question.