I'll be in the area around that time. Going on a 3 week roadtrip through 12 National Parks through South Dakota, Wyoming, Utah, and Colorado. My time in Utah last year wasn't enough, decided to outdo myself this year though. I don't know the exact dates that I'll be in that area, it's more likely that I'll have made it there sometime during the first week of June though.
I was under the impression that those roads were plowed and re-opened in May. I've seen quite a share of videos from soggy trails in June with massive walls of snow on either side of the road, but the roads were plowed and open nonetheless?