Are normal curvature invariant by local conformal maps?

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My idea: I thought this was not true because if I consider is true. Than I would contradict the result that says that every two regular surfaces are locally conformal. Thus if this was true, then all normal curvature would be locally the same. However I couldn't find a good and satisfactory counterexample or example to suport my idea. Would you help me with this? How to prove this question?