2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107762We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.10 pagesDifferential GeometryAnalysis of PDEs58J05;35J60text