User blog comment:Varzan Deodinm Beldinm/RWBY Episode 12... wow.../@comment-15145664-20131005004730/@comment-24038268-20131005012822

Glynda might doubt Jaune's abilities, but Ozpin's still the headmaster. If Ozpin says that Jaune will stay, Glynda can't do anything about it. She's not on the watch for Ruby's misbehaviour since Ozpin let her into the school, so she's probably pretty submissive to Ozpin's orders.

We don't know the actual size of the student/faculty body, but Jaune has the main protagonists as his allies (and that's always what matters). Even if they did believe Cardin, I think they would simply ignore the fact or continue to say that it's untrue/irrelevant whether they agreed with Jaune's actions or not. They might be disappointed in Jaune and lose a bit of faith in him, but they probably view Cardin as the greater evil/nemesis and argue with him because they can. A rumor spread by Team CRDL will probably hurt Jaune's reputation, but I think that JNPR/RWBY will go out of their way to fight it.