EPPA for finite partial Steiner triple systems

Determine whether finite partial Steiner triple systems have EPPA when partial automorphisms are required to extend only between closed substructures.

Background

The question isolates a natural class with a binary-function presentation and non-unary algebraicity. Closed substructures are those closed under completion of any triple containing two of their vertices.

References

Does the class of all finite partial Steiner triple systems have EPPA, where one only wants to extend partial automorphism between closed substructures?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question 13.3, subsection EPPA