Optimal path-decomposition conjecture for Johnson graphs

Let $F_k(K_n)$ be the $k$-token graph of the complete graph $K_n$, and let $\operatorname{pw}$ and $\operatorname{tw}$ denote pathwidth and treewidth. Optimal path-decomposition conjecture. The path decomposition of $F_k(K_n)$ constructed in the paper is optimal, and

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

The conjecture is based on computational verification for the case $k=3$ and is then formulated for general $k$. The source does not report a proof or disproof.