I would not expect a lot of snow around that time, but it might happen, that your get a little bit of snow in the northern part of your trip, but that would go away very soon. Most likely there will not be any snow on the highways, but you will find remainders of the snow from the winter beside the highway here and there, which is even possible in July.

I would not worry too much, pick good equipment for cold nights and rainy days and just go.