June was kind of a strange/interesting month for Julio. As you said, his BA for the month was very good (.289, not quite .300), but his OBP, SLG, and OPS were all pretty bad, and actually quite a bit below league average. I'm not bashing the guy, just stating the facts.
But regardless of of the exact numbers of any specific month, I think the overall point is that Julio once again had a pretty underwhelming 1st half. His overall numbers before the all-star break were slightly above league average (.252/.313/.417/.730). His numbers since the all-star break are very, very good (.292/.337/.569/.906).
It just gets a little frustrating seeing the same thing every year. We all know Julio is capable of being one of the best players in the league. We see it every year after the all-star break. It's just baffling to see a guy that is capable of playing so well, play so average for half a season every year.
Personally, I think Julio is capable of becoming a legit MVP candidate level player if he ever figures out how to hit for a full season like he hits in the 2nd half every year.