Minimum length of 2-uniformity networks

A 2-uniformity network is a network of lazy transpositions whose output permutation is uniformly distributed on every subset of two positions; let $U_2(n)$ denote the minimum length of such a network on $n$ positions. For $n\geq 2$, the conjecture is

$$U_2(n)=2n-3.$$

This is motivated by an explicit family of 2-uniformity networks of length $2n-3$; the conjecture asserts that no shorter sequences exist. Its status is left open in the source.