Is there a definitive source of information for whether or not it's open on any given day?
My guess is that this is something they won't know until much nearer the time - as it's so weather-dependent. 1 June is a date I found on a web site, but it was from a few years ago - so maybe that was the "target date" for that particular year. They may well be aiming to have it open for the Pentecost week-end (Pentecost is a movable feast), but my guess is that the final decision won't be taken until just before the day.
(I don't have any experience of the Stelvio Pass, but my family were regular users of the Timmelsjoch pass when I was a child.... and some years it was still closed as late as the beginning of July. And the Stelvio Pass is slightly higher than Timmelsjoch)