Johnson graph pathwidth conjecture

For integers $n$ and $k$ with $1\le k\le n-1$, let $F_k(K_n)$ be the $k$-token graph of the complete graph $K_n$, also known as the Johnson graph. The tree decomposition of this graph constructed in the paper is a candidate decomposition. Johnson graph pathwidth conjecture. The decomposition is optimal, and

$$\operatorname{pw}(F_k(K_n))=\operatorname{tw}(F_k(K_n)).$$

Moreover, the treewidth equals the upper bound established in the paper. The source gives no resolution of this conjecture; it is motivated by the authors' upper bound and computational evidence.