Regarding the hard route problem, how can we prove that fixing the meeting point of vertex x with the route will not cause double counting?

Hello everyone, in this hard route problem, I wonder if anyone could suggest or show me how to prove that fixing the meeting point between x and the route will not cause double counting, so that we can deduce that DP rerooting can be applied. Thank you all!

https://oj.uz/problem/view/IZhO17_road