Conjecture on the Bogomolov property for Galois representations attached to modular forms

Let $k\geq 2$, let $f\in S_k(\Gamma_0(N))$ be a normalized eigenform, let ${\mathbb{K}}_f$ be the number field generated by its Hecke eigenvalues, and let $\mathcal{O}_f$ be its ring of integers. For each finite place $v$ of ${\mathbb{K}}_f$, let

$$\rho_{f,v}:G_{\mathbb{Q}}\longrightarrow \mathrm{GL}_2(\mathcal{O}_{f,v})$$

be the associated Galois representation, and let

$$\rho_f=\prod_v\rho_{f,v}:G_{\mathbb{Q}}\longrightarrow \mathrm{GL}_2(\widehat{\mathcal{O}}_f).$$

A representation has property (B) when the field cut out by it has the Bogomolov property. The modular-form representation conjecture. For every normalized eigenform $f\in S_k(\Gamma_0(N))$, for every $v$, $\rho_{f,v}$ has property (B), and $\rho_f$ has property (B). The conjecture is presented as a generalization of Habegger's theorem; the source gives special cases, including CM modular forms, but does not state a general resolution.