I did a Army/New Army loop on June 28th, 2013. The hike was as Steve described it, with no real issue until I neared the top and encountered the snow covered trail. I could not see the trail on the other side (looking up), so I could not tell if the snow was atop the trail and was "solid" or if it might not cover some hazard, so I did up climb the last 50', which was (IMO) class 3 (exposure similar to the final 400 of MR).
After the up climb I did circle back down the trail to the snow, and could see my earlier position, and believe I could have crossed the snow. However, given what I "knew" the up climb made more sense to me.
Hard to say about this year. Seems likely the snow will be gone by July, but a late storm could change all that.