So this is something a little out of left field where Halo cannon is being told outside the realm of the universe. After our latest podcast, Bryan pointed out that an avatar item on the Xbox Marketplace has verified that Jun is still alive. The Commander Palmer Armor for your avatar has the following description:
Once an ODST, Sarah Palmer was hand-picked by former Spartan-III Jun A-266 to be one of the first candidates in the new Spartan IV program. Quickly proving herself during the defense of the UNSC Infinity from a hostile takeover by Insurrectionists, Palmer was soon called upon to lead a Spartan strike team during the historic Battle of Draetheus-V. Now the senior Spartan aboard Infinity, Commander Palmer went on to be instrumental in the recovery of the Master Chief and the defeat of Jul ‘Mdama at Requiem.
It’s doubtful that many people thought Jun was dead after Halo: Reach, but now we have confirmation.