Graph Algorithms · Superchallenge problem

A graph has vertices $1$–$6$ and weighted edges $12{:}2, 13{:}5, 23{:}3, 24{:}7, 34{:}4, 35{:}6, 45{:}1, 46{:}8, 56{:}2$. Its minimum spanning tree is $\{12, 23, 34, 45, 56\}$, weight $12$. What is the smallest possible total weight of a spanning tree that CANNOT be produced from this one by changing just a SINGLE edge – that is, one where at least two of the minimum tree's edges must be swapped out together?