I attempted the loop in late June about 15 years ago. The gate was still shut and locked near the top. You might call the agency in charge of the gate and see if they might have a estimated time once you get closer. I hear usually first part of July is a good time to expect it to be open.
Not sure if you are planning any camping in the area or not. The only advice I have for you is "DON'T!". Unless you like DEET loving, tent infiltrating, clouds of mosquitoes.....No...Really this is a training ground for the mosquito special forces. You have been warned.
On a lighter note, I might say do the loop, if you can, and then head down to Fields and get a hamburger and milkshake. You'll thank me later. Then head back up into the Alvord Desert to spend the night. Much less mosquitoes and a cool area in its own right.
If you want to spend the night on the West side of the Steens I recommend a time after the first frost so you don't have to deal with the mosquitoes.
Hey, did I mention mosquitoes.....