JUN become a member of spartan 4 program
It’s à mystery
Jun escaped on the second pelican sent out after getting the info/ core (unsure which one) from Dr. Halsey.
If you look into lore/books
Jun survived reach and was recruited to train the Spartan IVs and was one of the founding members; he approached Buck and recruited him for Fireteam Osiris.
He’s a recruiter/trainer for Spartan IVs. So you got your wish I guess.
Jun’s location and actions between the events on Reach and the Spartan-IV project are unknown. But the following below quoted from Jun-A266 | Halo Alpha | Fandom does shed some light.
> In January of 2553, Jun hand-picked Sarah Palmer, who was an ODST at the time, to become one of the first candidates for the SPARTAN-IV Program. Jun and Musa-096 began to train Palmer and the other new Spartans they recruited.
>
> While Jun and Lasky were giving the new Spartans a tour of UNSC Infinity, an Insurrectionist team led by Ilsa Zane took Infinity’s captain and bridge crew captive. In order to retake the ship, Palmer convinced Jun to allow her and the other SPARTAN-IV recruits to receive their armor despite that they hadn’t yet been cleared.
>
> While Palmer and Davis climbed Infinity’s outer hull to the Bridge, Jun, Lasky, and the other Spartans were engaged by a strike team of Insurrectionists. They held off the strike team long enough to seal the hallway doors, neutralizing the threat. When they finally arrived at the bridge, they found that Palmer and Davis had already neutralized Zane and secured the bridge.
Why did he have to bring Palmer into the Spartan IV program? 
Jun recruited buck into the spartan program
Maybe in Halo 6? aha
Jun is at home,waiting for that Halo 5 to be released on Tuesday with pack of mountain dew and a party bag of doritos.
I hope he has a cameo in Halo 5
You never know, Jun may come back in a cutscene or something. 343 likes Easter Eggs now too, so don’t rule out Jun’s resurrection in the games yet!
Jun became the recruiter for the spartan 4 project.
alive and well
became part of the S-IV
Alive for the most part
He became a Spartan-lV recruiter