Non-vanishing conjecture for the Rankin convolution of an Ikeda lift

Let $r$ be even and let $2n+r$ be a multiple of $8$. Choose an even unimodular matrix $S$ of size $2n+r$, and let $H(W)=F^{(r)}_f(W)$ be the Ikeda lift of $f$ to $Sp_{2r}$. For a Siegel cusp form $g$ of degree $r$, define

$$R\left(k+n+\frac{r-1}{2},H,\overline{g}\right)=\sum_{T\in \widetilde S_r(\mathbb Z)^+}\frac{a_{F^{(r)}_f}(T)\overline{a_g(T)}}{\epsilon(T)\det(T)^{k+n+\frac{r-1}{2}}}.$$

Non-vanishing conjecture. The Rankin convolution $R(k+n+\frac{r-1}{2},H,\bar g)$ is non-vanishing. The series is known to converge absolutely when $2n>r+2$, but its non-vanishing is not known.