Characterize interaction improvement via Böhm trees with possibly infinite η-reduction

Establish that the interaction improvement preorder Eint-imp coincides with the Böhm-tree preorder EB^{η↓}, defined as the variant of the extensional Böhm preorder (Definition 4.5) obtained by allowing possibly infinite η-reduction rather than η-equivalence; that is, prove Eint-imp = EB^{η↓}.

Background

The paper proves that the interaction preorder Eint equals the non-extensional Böhm-tree preorder Eg and presents chains of inclusions relating Eint-imp to extensional Böhm preorders. Building on these, the authors conjecture a precise Böhm-tree characterization for Eint-imp that switches from η-equivalence to possibly infinite η-reduction.

They note technical obstacles: η-reduction interacts subtly with interaction steps in the checkers calculus, and both rewriting methods and their multi-type semantics currently fail to manage η in this framework. Nonetheless, a refined Böhm-out technique already gives one side of the desired inclusion.