No, this does not hold. A rather handwaving counterexample is given below. I trust you can add the numbers, do the math, and formally verify this (I used this online solver to verify my claims).
Consider these points:
The top point is clearly far away, so it has to be connected to the closest points. The other links then follow, as shown here:
If we exclude the top point, it is more optimal to have the two top points connect to the center point, as shown below. So just short-cutting is not optimal: