I have not heard any news now a days. If you have link so please share so we could update ourself. I just googled to check if there anything happened but found nothing in fact found some incidents happened in 2013 and 2014 which I was not aware of before and I was there in end of August 2014 and also stay a night in Chillas and came through Babusar towards Naran but could not feel any security issue.
I am sure you will not find any difficulty in June to reach Khunjerab. More you can by calling on below numbers:
Hotel Riveria, Sost
Asif Kareem (Reception Boy)
Main bazar, Sost, Gojal Hunza
In June till Chillum you will find clear road and in Deosai you may find muddy track due to melting snow. But you can confirm about route from Astore or Chillum. If you crossed Deosai, you will not find any problem to reach Skardu and Gilgit-Skardu road is a narrow road and due to rain some parts are land sliding areas and no one knows when road goes block due to large stones.
Babusar should be open in June i guess.
I would recommend you to plan your tour in late June or mid July to get access to all areas.