Theorem 1. There is an $O(n\log n)$-time algorithm for the problem in the post.
Proof. We first state two utility lemmas, for an arbitrary edge-weighted graph $G$. We postpone their proofs, which are standard, to the end.
Here is the first lemma. Most likely this is already in the Monma and Suri paper.
Given any bipartition $(C_1, C_2)$ of $G$,
let $W(C_1, ...