BUG REPORT: Universal Control and typing in login password after computer wakes

here are the computers specs and the scenario:

Mac Studio 2022 M1 Ultra - Ventura 13.6.6

MacBook Pro 15" 2016 - 12.7.5

• Mac Studio initiates Universal Control with the MacBook Pro; using Mac keyboard and trackpad. Mac Studio screen locks and screen saver starts; MacBook Pro is set to NOT auto-sleep or auto-lock - it's set to stay on always.
• I come back to my Mac Studio after 1 hour; with the screen saver still on I press a key to bring up the login screen, it comes up and so naturally I start typing in my password; since this is such a natural occurrence I do it nonchalantly without looking at the screen and press enter.
• BUT WAIT, nothing showed up in the password input field... what happened the password I just typed? Oh look, my keyboard is actually still targeted to the MacBook Pro, and I just inadvertently typed my password into a chat window and sent it for all to see...
• But why did my Mac Studio react with popping up the password input field when I pressed my keyboard? Naturally I would start typing when that happens... I wouldn't know that my keyboard was still targeted to a chat window on the "secondary" computer.

Quite deceiving that the same keyboard would make my Mac Studio react and wake up from the screen saver and show me the password input field when it was actually targeting the chat field on my MacBook Pro.

I feel the keyboard needs to be locked to the "primary" computer after an auto-lock or sleep is activated from said computer. And when it wakes, the keyboard should be targeted back to the "primary" computer even if it was left on the "secondary" computer. Or perhaps Universal Control should be disabled altogether once any computer initiates a screen saver or auto-lock.