Nonminimal-coupling monopole static Love-number vanishing

Prove whether the static logarithmic Love number [?] for the monopole of a non-minimally coupled scalar field vanishes on any static, spherically symmetric, asymptotically flat background, extending the established result for a minimally coupled scalar field.

Background

The paper notes that the leading anomalous behavior for scalar monopole perturbations simplifies when the static logarithmic Love number vanishes. For a minimally coupled scalar field on any static and spherically symmetric background, the paper states that this vanishing is known. It then suggests that the result may persist for non-minimal couplings, but explicitly acknowledges that a complete proof is unavailable.

References

Presumably this remains true upon including non-minimal couplings, although we are not aware of a full proof.

— The Love for Three Logarithms: Universal Running of Dynamical Love Numbers  (2610.01152 - Garcia-Saenz, 1 Oct 2026) in Section 2.2, paragraph discussing the [?] case; footnote following Eq. (MST discussion)