Sabatier conjectures
OK, here's a big hint.

Take any $G,A,S$ that satisfy the premises of the question.  Now add a new vertex $v'$ to the graph, and draw an edge from $v'$ to one of the vertices of $A$ with a weight of your choice.  (This means that $v'$ will be connected to the rest of the graph only by this one new edge; there are no other edges out of $v'$.)

What can you say about this new edge?  Does it cross the cut?  Is it a light edge?  Is it a safe edge?  Does this suggest any method for choosing the weight on that edge, to make things fortuitous to you?  Hint: if you think about it (or try some examples), I bet you'll see how you can arrange things to get a violation of the conjecture.

People learn best by finding the solution for themselves (rather than reading someone else's solution), so I'm not going to post more than a hint.  This hint ought to be enough to understand what's going on here....