I'd bet so. Here's an interesting visualization (that also does not bode well for wildfire season):
Big Sandy, Deep Lake, Temple Mtn. area and beyond, June 1, 2020:
Same area, June 1, 2019:
Here's the closest I could find for a date in 2019 that looks like it does now. June 29, 2019:

It lacks the lower elevation lighter amounts of snow, but the higher elevation, more significant, lingering snowpack is the similar, maybe even higher in some places. So we're probably 20-30ish days ahead of schedule as it stands right now.
So since your trip begins 16 days from now, hypothetically, here's 16 days after the similar snowpack day of June 29, 2019 (which happens to coincide with the last day of my trip in the area last year:
Of course, temperatures may be a little lower the next 16 days than they were in early July 2019, just because we're in June instead of July (debatable though with how hot it's been around the mountain west the past week). But I'm almost certain you'd be able to get to Deep Lake with no issues in 2-3 weeks barring wild swings in weather. I'm also certain there will be wild amounts of mosquitos. Haha.
Source:
https://www.nohrsc.noaa.gov/interactive/html/map.html