5 problems

Complete-graph extremality conjecture. For every graph $G$ with $n$ vertices,

Spanning-subtree probability conjecture. For every graph $G$ with $n$ vertices,

Almost-spanning subtree ratio conjecture. For every graph $G$ of order $n$,

Connected-graph contraction conjecture. Contracting any edge should reduce the mean subtree order by at least $\frac13$, with equality only for a path:

Mean subtree order extremality conjecture.