Addendum:βϕ can be obtained from βμνg using the contracted Bianchi identity. The
conformal anomaly appears precisely as the integration constant. This mechanism is very
reminiscent of what happens in unimodular gravity theories. These are theories with
invariance only under volume-preserving diffeomorphsisms. They were first considered
by Einstein in Ref. [4] and more recently by various authors in Refs. [2, 5, 7, 6]. See
Ref. [1] for a very recent paper with many references.