Okay, I know I shouldn't, but I'm playing the eyeball game again. Here's the volume decline during June:
And here are the SAT and SLP maps for both halves of June:
The first half of June looks very similar to that of 2013 wrt both SAT and SLP, and maybe just a little like 2015. They both lost around 2800 km3.
For the second half of June this year again looks most similar to 2013, and there are some similarities with 2012 and 2015. The average of those three years is 3600 km3.
That makes the total 6400 km3, but I'm going to add 100 km3, because of the heat wave over the ESS at the start of the month (where the ice is supposed to be really thick this year).
So, my guess is that total June volume loss will come in at around 6500 km3, plusminus 100km3. That would take 2016 into second place, around 500 km2 higher than 2012.
I wouldn't be surprised if June total volume loss is lower, but this is what my eyeballs are saying.