Its not a common problem, so I am just suggesting you try standard troubleshooting. You could try repairing or reinstalling the the browser you use if this does not help.

1. temporarily disable sleep under power settings (set sleep to never)
2. Open windows control panel and select Security and Maintenance. Click on Maintenance. Find the Start Maintenance button and click it. Takes maybe 30 minutes or sometimes more depending. Leave the Maintenance window open so you can see when it is finished. You can also monitor the system with task manager or resource monitor.
3. After that is complete restart the computer and check the issue.
Reverse any change you made to the power sleep settings.

is to check your browser start page settings.

It wouldn't automatically close on log-off or on closing the main DU page(s).

Since it is just doing one function that is not tied to your session, it seems fairly harmless to either leave open or just close it.

I'd try going into your DU settings and turning spell check on then off again.

Illogical, I know, but I do a similar thing with my browser add ons when they play up and it works like a reset.

I went into DU settings but couldn't find spell check. Out of frustration, I changed all the settings and then changed them back. That seemed to work.