Enriched profunctor example for univalent weak double categories

An univalent weak double category is a weak double category whose corresponding double bicategory is univalent. Its objects, horizontal morphisms, vertical morphisms, and squares may be interpreted respectively as enriched categories, enriched functors, and enriched profunctors.

Enriched profunctor conjecture. (Enriched) categories, (enriched) functors and (enriched) profunctors assemble into a univalent weak double category, but not into a univalent double category.

This conjectural example is the categorified analogue of the double category of spans and shows that the weak, symmetric notion is genuinely broader than the strict univalent double-category notion.