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This is a the MIUI ROM based on JB 4.1.2.  However, the problem with the widget not updating precedes the use of this ROM.  Even when I was on stock ROM (ICS 4.0.4) the widget was not updating. I only had this ROM for a couple of weeks.  I have had a paid version of Gmail Widgets at least for four months, even before I had the Galaxy SIII.  I was using the widget already on my Galaxy SI with Android 2.3.  Even then the updating of the widget was not consistent.  I only reported this now since I finally had the time to provide the feedback and see if this can be fixed.

I waited at least an hour.  I have attached screen captures to illustrate.  The first screen capture of 10:48 shows in the status bar that I have received messages through Gmail with the Gmail icon.  The widget however does not show any new items and was not refreshed since 08:32.  Once I manually refresh the widget shows that I have three new messages and the first one arrived at 09:27 second screen capture of 10:49).  All new messages are arriving in the standard inbox.  I have a Samsung Galaxy SIII and it is rooted.