Bogomolov property conjecture for modular Galois representations

Let $f\in S_k(\Gamma_1(N))$ be a normalized eigenform, and let $\rho_{f,v}$ denote its Galois representation at a non-Archimedean place $v$ and $\widehat{\rho}_f$ the product of these representations. For a representation, property (B) means that the field fixed by its kernel has the Bogomolov property. Modular representation conjecture. Then:

$$\text{(a) for every non-Archimedean place }v,\ \rho_{f,v}\text{ has property (B);}$$ $$\text{(b) }\widehat{\rho}_f\text{ has property (B).}$$

The conjecture is motivated by Habegger's theorem and known results for modular representations; part (b) implies part (a), but the converse is not immediate because property (B) is not preserved under composita. Both assertions remain open according to the supplied status information.