Mazur's non-vanishing conjecture for Rankin–Selberg L-values

Let $f$ be a cuspidal newform of weight $2$ on $\Gamma_0(N)$, let $K$ be an imaginary quadratic field of discriminant prime to $N$, and let $\chi$ be a finite-order Hecke character of $K$. Write $L(s,f,\chi)$ for the associated Rankin–Selberg $L$-series, and let $\chi_0$ be the trivial character. Fix a prime number $\ell$ prime to $N\operatorname{Disc}(K/\mathbb{Q})$. The sign $e(\chi_0)$ in the functional equation distinguishes the definite and indefinite cases.

Mazur's conjecture. If $e(\chi_0)=+1$ (the definite case), then

$$L(1,f,\chi)\ne0$$

for all but finitely many characters $\chi$ of $\ell$-power-order conductor. If $e(\chi_0)=-1$ (the indefinite case), then

$$L'(s,f,\chi)\ne0$$

for all but finitely many characters $\chi$ of $\ell$-power-order conductor.

This conjecture predicts non-vanishing of central Rankin–Selberg values in the definite case and of the relevant derivatives in the indefinite case, as the characters vary through those of $\ell$-power-order conductor. The supplied text does not state whether it has been resolved.