TomH20, the snow that fell was not heavy, judging by the webcam views available. You will likely encounter snow on the switchbacks, but so many people are already going that it will be packed down. Taking microspikes would likely give you the added traction to help over the packed snow.

Ernest, 6/13 is weeks away. You will likely encounter pretty easy-going conditions, but likely some patches of snow to cross. Depending on your comfort level, microspikes might help.