Bounding the signed count of real bitangents to plane quartics

Using methods from enriched enumerative geometry, Larson and Vogt gave a signed count of the number of real bitangents to real smooth plane quartics. This signed count depends on a choice of a distinguished line. Larson and Vogt proved that this signed count is bounded below by 0, and they conjectured that the signed count is bounded above by 8. We prove this conjecture using real algebraic geometry, plane geometry, and some properties of convex sets.