The oriented swap process–last passage percolation distributional identity

Let $\bm{U}_n$ denote the vector of oriented swap process finishing times and let $\bm{V}_n$ denote the point-to-line last passage percolation vector. The oriented swap process–LPP conjecture. For all $n\geq 2$,

$$\bm{U}_n \overset{D}{=} \bm{V}_n.$$

The equality is conjectural and remains mysterious; the paper gives an algebraic-combinatorial reformulation and proves related marginal identities, but does not establish the full vector-valued distributional identity.