Existence of an interesting ff-protomodular enriched category

Determine whether there exists an interesting preorder-enriched category that is ff-protomodular, where ff-monomorphisms are the full and faithful monomorphisms.

Background

The paper introduces a relative form of protomodularity in which change-of-base functors are required to be conservative only with respect to a specified class of morphisms. In the enriched setting considered near the end of the paper, the relevant class is that of full and faithful (ff) monomorphisms.

The authors observe that, in the examples studied, ff-monomorphisms are not sufficiently strong to yield relative protomodularity. This motivates the unresolved question of whether any interesting category enriched over preorders satisfies the ff-monomorphism version of protomodularity.

References

Hence it is an open question whether there is an interesting $$-protomodular $$-category for $$ the class of ff-monomorphisms.

— How protomodular is your favourite category?  (2610.09739 - Clementino et al., 7 Oct 2026) in Introduction, final paragraph before Section 1

We do not have an example of an $$-protomodular category which is not protomodular.

— How protomodular is your favourite category?  (2610.09739 - Clementino et al., 7 Oct 2026) in Remark following Example 5, Section 5 (the section titled “$$-protomodular categories”)