Nonvanishing of Rankin–Selberg central values for ring class characters

Let $f$ be the eigenform, $K$ the quadratic field, and let $\mathfrak{X}$ denote the relevant set of characters. Write a ring class character as $\rho=\rho_0\rho_w$, of any given $p$-power ring class conductor $c$, and let $\psi=\chi\circ\operatorname{N}$ be a cyclotomic character in $\mathfrak{X}$. Nonvanishing conjecture. For every such $\rho$, there exists a cyclotomic character $\psi=\chi\circ\operatorname{N}$ in $\mathfrak{X}$ such that

$$L(1/2,f\times\rho\psi)\ne 0.$$

This criterion is intended to establish the remaining generic cases of the paper's nonvanishing theorem, using the Weierstrass preparation theorem; the source does not state whether the conjecture has been resolved.