EPPA for finite partial binary algebras

Determine whether the class of all finite structures in a language with one partial binary function has EPPA.

Background

This is a basic example outside the scope of the general EPPA theorem because it contains non-unary algebraicity.

References

Let $L$ be the language consisting of a single partial binary function and let $\mathcal C$ be the class of all finite $L$-structures. Does $\mathcal C$ have EPPA?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question 13.2, subsection EPPA