Graph Algorithms · Superchallenge problem

A graph has vertices $1$–$6$ and weighted edges $12{:}3, 13{:}8, 23{:}2, 24{:}6, 34{:}5, 35{:}4, 45{:}7, 46{:}9, 56{:}1$. Its minimum spanning tree includes edge $34$ (currently weight $5$). If edge $34$'s weight were increased, what is the largest value it could reach while the graph STILL has a UNIQUE minimum spanning tree that includes it (i.e. the value at which a tie first appears)?