The nonvanishing conjecture for quaternionic theta lifts of Hilbert newforms

Let $F$ be a totally real number field, let $\boldsymbol{c}$ be an odd square-free level, let $g\in S_{\mathbf{2}}(\boldsymbol{c})$ be a newform, and let $v_g$ be a Hecke eigenvector associated with $g$ in the quaternionic setting. Let $B$ be a quaternion algebra and let $\theta(v_g)$ denote the associated theta lift. For each prime $\mathfrak{p}\mid\boldsymbol{c}$, let $W_\mathfrak{p}$ be the Atkin–Lehner involution and let $w_\mathfrak{p}$ be its eigenvalue on $g$. The nonvanishing conjecture. The form $\theta(v_g)$ is nonzero if and only if $L(g,1)\neq0$ and $B$ ramifies exactly at the archimedean primes and at all primes $\mathfrak{p}\mid\boldsymbol{c}$ for which $w_\mathfrak{p}=-1$. This is presented as a naive Hilbert-modular generalization of a result of Böcherer and Schulze-Pillot for classical modular forms. The supplied text gives no resolution of the conjecture; it is supported there by a later numerical verification in one example.