Mac Studio Display sleeps when nearby iMac does

I switched from my iMac to a Mac Studio M1 with a Studio Display. I did not migrate my user account from the iMac. I created a user ID on the Mac Studio, then migrated selected items from the iMac to the Studio. Here's the problem: the Studio Monitor goes to sleep and tapping the keyboard does not wake it. However, tapping the keys on the nearby iMac does. If I shutdown the iMac, the Studio display will not come on. Before you suggest it, we've gone through all the preferences in Display, Energy Saver, etc. We've talked twice to Apple Support. We reinstalled OS Sonoma. Nothing so far has worked. We suspect it's the Studio Display itself is the problem