Outlook update of June to version 2306 removed the startup options that were there in the prior version (2305). Those startup options about opening items that were left open when Outlook last closed are Always open or Never open or Ask to open. The choice you selected was stored in the registry key above (as values 0, 1, or 2) that people are adding to the 2306 to fix these Ask boxes when Outlook starts up. Basically the registry key from the 2305 version is used to "fix" this problem in the 2306 version. I never added that Registry key, but when I reverted to the prior 2305 version that reversion added that key to my Registry as well as restored the start up options in Outlook settings which the value of that key changed as I changed the start up options in Outlook's settings. Even that registry key being added to the 2306 version doesn't seem to "fix" this problem all the time and appears that removal of the start options does not have Outlook looking for those options at all and the problem is hit and miss no matter what the key's value is set to. If you don't leave any emails in the Inbox when you close, then Outlook doesn't see any emails to say it was opened when it is closed and thus no question box.
The only sure way to get rid of these pop up boxes is to:
Stop Outlook updates
Revert back to the 2305 version
Wait until you hear that Microsoft has fixed this glitch and restart updates.
For anyone who has the same issue.
This is how to revert back to Build 2305:
First, you need to disable Office updates. If you don’t disable updates, the problem update will reinstall itself. You can disable updates from the File menu in any Office application.
In Outlook go to File, Office Account, Update Options and click onDisable Updates.
Open the command prompt as administrator. If you are using Outlook 2016, 2019, or 365, copy the following line, then right click in the command prompt window to paste it. "C:\Program Files\Common Files\microsoft shared\ClickToRun\officec2rclient.exe" /update user updatetoversion=16.0.16501.20210
INCLUDE THE "QUOTES" AS SHOWN ABOVE. If you look at the screenshot below you'll see the first time I entered the command I forgot the beginning quote and it was not recognized. I then put in the quote, pressed Enter and the "update", or in this case "reversion", downloaded and installed.

To revert back to any other build, you can find the build numbers here:
Version numbers of update branches for Office 365 clients.
The format to revert back to a prior build is:
"C:\Program Files\Common Files\microsoft shared\ClickToRun\officec2rclient.exe" /update user updatetoversion=16.0.xxxxx.yyyyy (where you change xxxxx.yyyyy to the build you want)
I just did it and it works without any issues. I hope this helps others.