Answered by Shaykh Faraz Rabbani
If one walks into the university prayer room or mosque and has not prayed the 4 sunnah of dhuhr, should he join the congregation if it is already in progress?

Walaikum assalam,
1. Definitely. One should join the congregation, for it is more strongly emphasized than the pre-fard sunna prayer, and then pray the pre-fard sunna after the obligatory prayer, except at Fajr and Asr times. At these two times, it is not permitted to pray any non-obligatory prayers after the obligatory one. [Shurunbulali, Imdad al-Fattah Sharh Nur al-Idah; Ibn Abidin, Radd al-Muhtar]
2. At Fajr time, however, one should pray the pre-fard sunna even if the congregation is in progress, but only if sure that one will catch the congregation before the imam’s final salams. [ibid.]
And Allah alone gives success.
Wassalam,
Faraz Rabbani