Do any two spanning trees of a simple graph always have some common edges?

I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?

No, consider the complete graph $K_4$:

It has the following edge-disjoint spanning trees:

StackExchange Computer Science Q#101038, answer score: 47