Even if the edge weights are unique, Can't it have multiple spanning trees? Like if ST1 contains two edges with weights 4 and 3, and ST2 contains the same edges as ST1 but instead of those two edges it contains edges with weights 2 and 5 (both pairs summing up to 7). Both will have the same total edge weight sum.
I am not sure about this, is there anything wrong in my approach?