It's not empathy when a person is passed over for a role because of their skin color due to unwritten HR policies, and people using the word "woke" are generally referencing the unethical and illegal aspects of such policies.
So much for "having empathy for someone outside my identity group".
Given that many such comments also amount to gaslighting, that's what bothers people about the whole idea. It was a good experiment, but it's time to end the policies and the attitude it brings. There are better things to argue over, this isn't one of them.