It sure is already possible to do the Ala Kol trek in the second week of June!
There might still be some snow at some places, so good waterproof shoes are necessary and also warm clothes, as the nights will still be cold at high altitude.
There are yurt camping options on this trek, which makes the evenings a bit more comfortable. If you're camping in tents, be sure to have a sleeping bag that is rated to at least 0 degrees C, but ideally for below freezing.
