Hollywood:
Can't say for White proper, but I can tell you the east side of the Whites (where the final switchers of the jeep road take you to the summit) has a good amount of snow on it. White is on the far left of the photo in the distance, this pic taken from the summit of Boundary last Saturday.
Snow on Whites There are a lot of sections of the long road in that are north-facing as well, so they will also keep snow. It's a little deceiving looking up from B-town, as the west is melted off entirely.
The final slope isn't that steep, so I would think a bit of snow would actually make it more pleasant! The deepest I sank into drifts further north last weekend was about knee-deep.
Hope it works out for you! Agree with Steve: call the White Mtn Ranger station first.
-L
