Bad news: Adorama and B&H Photo will both be closed next week for Shavuos (June 11 through June 13). That means that there is a low chance (but not impossible) for Nikon to introduce a new product (Z6III) while the two major US stores are closed. In the past two decades, I do not ever remember Nikon announcing a new product while Adorama and B&H are closed (please correct me if I am wrong). Stay tuned for updates.
Update: I was wrong, Nikon announced the AF-S 28mm F/1.4E ED, a few DX lenses, and the Coolpix W300 during Shavuot 2017 (thanks RC Jenkins). In that case, we are back on track for the Z6III to be announced next week.