Incognito mode and tabs placed 'on hold'

Personal device, connected to work network. Let's assume that Chrome has been used in the past 24 hours, but isn't currently open. (I don't know whether that means it counts as a running application, but it feels like that matters.) So the first question is, what if I just never use a browser on the work network all day? Does browsing information even matter then? I know my tablet connects to the network pretty much right away when I walk in. Does it also immediately fetch the web pages on my browser?

But let's say I do open up Chrome. The top tab pulls up some article about narwhals or the guy who performed his own appendectomy at the Antarctic base in 1962, or whatever bullshit I was curious about last night. Not work-related, but also not too embarrassing. Meanwhile, there could be Incognito Mode tabs also not closed... but the browser is no longer in Incognito Mode. So I don't know if they matter.

So basically, if I do open Chrome at work, does the network also try and fetch or update the tabs over there in Incognito mode?