If $A$ and $B$ are $2\times 2$ matrices such that $A^2 - B^2$ is invertible, is $A-B$ invertible?

Question

Let $A$ and $B$ be $2\times 2$ matrices such that $A^2 - B^2$ is invertible. Is $A-B$ necessarily invertible?

This doesn't seem like it should be difficult but I just can't come up with a solution.

Answer 1

The answer is no. For instance, consider $$ A = \pmatrix{0&1\\1&0}, \quad B = \pmatrix{0&1\\0&0}. $$