Let $A$ and $B$ be positive-semidefinite matrices with norm satisfying $\|A\| \leq 1$ and $\|B\| \leq 1$. Is the following inequality true?
$\|\mathbb 1 -AB-\sqrt{\mathbb 1-A^2}\sqrt{\mathbb 1-B^2}\| \le \| A-B \|$, where $\mathbb 1$ is the identity matrix.
Here, $\|\cdot\|$ denotes the operator norm. (Insights into the applicability of this inequality under other unitary-invariant norms would be valuable for me.)
In the case where the inequality does not hold generally, is it possible to introduce an additional factor or set of conditions that would make it true?
I have already been able to establish the validity of the inequality when the matrices commute. Thanks.