As of 5/25 Dollarhide Summit, west of Ketchum on the Main Route, is still snowed in. It is predicted it won't be open until late June. It is advised that trips begin no earlier than the 3rd or 4th week in June and even then expect to encounter snow on the ground.
This is directly from the local source:
And to be clear I don't mean it would be difficult, I mean it would be a dangerous, wet, cold snow shoe hike for miles and miles. The danger part is from an unstable snow pack (snow that is moving and sliding as it melts) , isothermic snow pack that is bottomless (even with snow shoes you would sink to your waist). As the snow continues to melt the creeks and rivers are running deep and they too are dangerous. That also means that many of the hot springs are underwater and inaccessible.
While the online 'snotel' websites may be good for reporting overall conditions in the region, they may not be representative of what the road conditions are specifically.
If you have further information, please post it here.