Mail not refreshing

I'm running OSX 10.11.6 with Mail 9.3 (3124) on a 2009 Mac Pro. About a week ago Mail started to hang up on "Downloading Messages". When I first start Mail it retrieves messages but then will not retrieve more. I have to quit and restart Mail to get new messages. Otherwise everything seems to work fine. I'm connecting to Office365 and Gmail, but it only seems to be freezing on Office365 retrieval.

I'm seeing the identical problem on my Macbook Pro, also running OSX 10.11.

I saw some similar complaints with a newer version of OSX (10.12 or .13) that was reported resolved last spring. Unfortunately I can't update past 10.11 due to having a 2009 Mac Pro (Macbook Pro is also a 2009 marooned on 10.11). Has anyone else been experiencing this in 10.11? I REALLY don't want to have to switch to using Outlook.

Thanks,

Michael