Low-dimensional flag versus quadratic triangulations

Determine whether there exists a lattice polytope of dimension at most four that has a flag unimodular triangulation but no quadratic triangulation.

Background

The census shows that among (0,1)-polytopes, the first examples with flag unimodular triangulations but no quadratic triangulations occur in dimension five. The problem asks whether allowing arbitrary lattice polytopes produces such an example in dimension four or lower.

References

Is there a lattice polytope of dimension at most four that has a flag unimodular triangulation but no quadratic triangulation?

— Most $(0,1)$-polytopes are not normal  (2609.02778 - Morales, 2 Sep 2026) in Unnamed Problem following Problem Koszulness in a family of (0,1)-polytopes, Section 5, Open problems