Users opening multiple shared mailboxes results in poor performance in Outlook Desktop

Thanks for the response Vasil. 

Is there a definitive way to establish that Outlook is having trouble with this configuration. I mean I personally agree and suspected this wasn't a good setup, but the customer insists that when they were using on-premise they never had these issues. There's obvious reasons why they may have had better performance but I'd like to be able to nail down the exact reason why Exchange Online is not going to offer them the same performance. 

I just tested SharePoint site mailboxes but the performance from Outlook 2016 client would make this a bad choice. It's handy in that you see the mailbox in Outlook client as well as the documents library, but ironically you can't move files around in the Documents folder from Outlook. I've read elsewhere that you should also see the site mailbox in OWA but so far I don't see it. So I'll have to investigate that further as well. Also, in Outlook client if you right click a document in the Documents Folder and click forward you get the error "The operation failed. The messaging interfaces have returned an unknown error. ...". You can't send on behalf of the site mailbox either and when you create a new email, albeit from your own address, you can attach anything from the SharePoint site's document library, so I don't see the usefullness of the documents showing in Outlook. 

Feels to me like so many of the collaborative features MS is putting into the Office 365 platform simply don't work or require more configuration to make them work and sometimes that doesn't even help. 

Do you have a link where I can see how to configure or control the cache as you suggested?