The correspondence between Green's relations and fine equivalence relations

Let $U$ be a subsemigroup of a semigroup $S$. Green's relations are equivalence relations on $U$, while the fine equivalence relations are equivalence relations on $S$ obtained from the prolonged images of relations on subsemigroups.

Correspondence conjecture. The Green's relations on a subsemigroup $U$ of $S$ have as counterparts prolonged images in $S$, namely the fine equivalence relations on $S$.

This proposed correspondence is intended to give a deeper interpretation of the fine equivalence relations developed in the paper. The supplied text does not state whether the claim has been proved or remains open.