Update: PayPal has confirmed the start of the Steam Summer Sale: June 23. I don't know when PayPal became the official herald, but it has leaked sale dates so often in the past that I presume Valve just gave up and put it in charge.
It’s official: the @Steam_games sale starts 23rd June. https://t.co/6EdbWFScCWJune 16, 2016
That word comes from an unknown developer who seems to have little interest in the finer points of confidentiality, who posted with the info. Oddly enough, the message is in english, but the dates themselves are in Russian, suggesting that the leak either originated there, or is fake.
Obviously this is all entirely unofficial and unverified, so take it for what it's worth, but also bear in mind that basically the same thing —the Steam Summer Sale dates were leaked onto Russian social network , by the same guy (Vyacheslav “MrFreemanBBQ” Konchits) who posted this year's dates—and proved out to be accurate. I've reached out to Valve for confirmation of the dates, and I'll let you know if I get anything back.