A--B--C, where AB weight = 2, BC wight = 2 and AC weight = 5. Shortest path from A--C is via B with weight 4. Now if we add a constant weight say 3 to each of them, shortest path becomes A--C with weight 8.
If edge weights are distinct MST remains same if a constant weight is added to all edges. Otherwise the output of a MSTalgorithm can change as multiple solutions are possible for the MST problem. Still the old solution is valid (correct) and hence the statement is true.