After receiving an email from a friend explaining that PPP might apply also to self-employed individuals with no employees, I asked another friend to research the veracity and get me info.
The second friend told me that he applied on behalf of the school he works for, and gave me the rundown of what seems to get approved (with big banks it will mostly be customers with a valuable relationship, with small banks it might depend on completeness of the application).
He advised me that if I want a chance at round 2, I better act fast and make sure not to apply with a large bank. The challenge is finding any bank that will even allow you to submit an application now, so it's ready for SBA submission once there's a reopening.