Uniqueness of tournament-kernel representatives up to measure-preserving transformations

Let $W$ and $W'$ be equivalent tournament kernels, meaning that they represent the same tournament limit. A measure-preserving transformation is a map from $[0,1]$ to $[0,1]$ preserving Lebesgue measure. Tournament-kernel equivalence conjecture. If $W$ and $W'$ are equivalent tournament kernels, then there exist measure-preserving transformations $\sigma_1,\sigma_2:[0,1]\to[0,1]$ such that

$$W(\sigma_1(x),\sigma_1(y))=W(\sigma_2(x),\sigma_2(y))$$

almost everywhere. This would give a canonical description of tournament limits up to measure-preserving changes of coordinates, resolving the non-uniqueness of tournament-kernel representatives; the source states that no proof is known in the literature.