A determinant inequality for all nonnegative base powers

For positive definite matrices $A, B$, every $k \ge 0$ and $0 \le p \le 2$, we
prove
$$\det(A^k + |AB|^p) \ge \det(A^k + A^pB^p).$$
In fact, a log-majorization of the corresponding normalized matrices holds:
$\lambda(A^{(p-k)/2} B^p A^{(p-k)/2}) \prec_{\log} \lambda(A^{-k/2} |AB|^p A^{-k/2})$.
The previously proved range $k \ge 2$ is due to Ghabries, Abbas, Mourad and
Assi. We treat the remaining base powers using a normalized order implication.
Two applications of the Furuta inequality establish complementary parameter
regions. The identity $\bigl| |AB|^{-1} B \bigr| = A^{-1}$ interchanges the two
parameters and closes the remaining range.