Should warm and cool grey tones be used together in UIs?

I am currently working on redesigning a web app and want to reduce the amount of colors used, espcially the lot of different grey tones - warm and cool mixed up as accents for UI elements and the background.

I am wondering, wether it's a good idea to mix warm and cool grey tones in the UI of an application (web or desktop) and if so, how similar are they allowed to be? Or is there a best practice? Especially in combination with a small set of accent colors?

Personally, I feel like it is a bad idea - as different warm/cool grey tones (both used as 'background accents' somehow don't go too well together in my aesthetics. But as a designer I don't just want to rely on personal opinions. So, are there any rules of thumb or guidelines to follow?