F1 or J1?

Thank you in advance for any help!

My hu*****and's OPT (F1) will be ended on July 13. His university wanted him to work as a visiting scholar and applied for J1 visa for him, which will start on July 12. However, the I-797A notice we received last week says the valid date is from 06/13. So does that mean he has been in a J1 status? Is it possible to cancel this status and keep his F1 stutas until July 13? Or does he need a J1 waiver if he has had a job offer?