As is typical this time of year, most areas that are shaded are still going to have some snow. Right now, there is snow down to about 6,500 ft on the steeper north facing (sun protected) slopes around the area.
So yes, I'd expect some snow on a good part of the N. Backbone trail, especially from Pine Mtn over to Blue Ridge as I think the sun angle hits pretty low on a good part of that section of the trail.