Barron urged Council to reject the proposed guidelines, which currently exclude Penn State employees from being selected as Mayor Don Hahn's replacement.
"He's a fifth-year senior, I want to reward him for his loyalty."
Cain got his first collegiate start against Michigan State on October 26, but he was listed as the first running back on the depth chart for the first time this season.
The Nittany Lions dropped one spot from their previous ranking of No. 7. They previously held the No. 8 spot for three weeks earlier in the season.
Both of Penn State's national championship victories cracked the list's top 10, and the team's 52-49 Rose Bowl loss to USC in 2017 was ranked as the 24th-best game in college football history.
The Nittany Lions moved up after sweeping Wisconsin by a combined final score of 10-3 over their two-game series.