Dual citizens and EES registration

Hello! An older family member who is a dual US/Italian citizen accidentally left Italy recently on her US passport, and registered herself in the EES system without realizing what she was doing.

When she returned to Italy a few days later on her Italian passport, she explained the issue and asked them to remove her from the EES system, and they said there was no problem and wouldn't remove it, although I believe they are supposed to if asked.

Timeline:

April - entered Italy on Italian passport

June 5 - left Italy on US passport

June 8 - entered Italy on Italian passport

June 15 - needs to leave Italy again for several months

As she's registered in EES on her American passport, should she leave Italy again on the US passport just so she isn't registered as an over-stayer? Or should she try and get removed from the EES system again?

Any suggestions would be appreciated!