Position of EC, ES, and ER in the analytic equivalence relation hierarchy

Determine the position, with respect to Borel reducibility among analytic equivalence relations, of the equivalence relations EC, ES, and ER defined as follows: EC is the equivalence relation on pairs of closed graphs G,H on 2^ω given by (2^ω,G) =_≤2 (2^ω,H); ES is the equivalence relation on pairs of homeomorphisms f,g in H(2^ω) given by (2^ω,G_f) =_≤2 (2^ω,G_g); and ER is the equivalence relation on pairs of graphs G,H on a fixed countable dense subset D ⊂ 2^ω given by (2^ω,G) =_≤2 (2^ω,H). Ascertain their exact classification-theoretic complexity within the Borel reducibility hierarchy of analytic equivalence relations.

Background

In Section 6 the authors introduce several equivalence relations associated with continuous injective homomorphisms between graphs or graph-like structures on compact zero-dimensional spaces. They prove that flip-conjugacy on minimal homeomorphisms (FCO) is Borel reducible to these relations and conclude that, as sets, these relations are analytic complete.

Despite these completeness results as sets, the precise status of these relations as analytic equivalence relations under Borel reducibility is not determined. The authors therefore pose a question about locating these relations—EC on closed graphs on 2^ω, ES on homeomorphisms via their induced graphs, and ER on graphs on a fixed countable dense subset of 2^ω—within the established hierarchy of analytic equivalence relations, in light of known benchmarks such as the complexity of conjugacy on H(2^ω) and bi-homomorphism on countable graphs.