Nonvanishing conjecture for the theta lift from quaternionic modular forms

Let $p$ be a prime, let $\widetilde{\mathscr{O}}$ and $\mathscr{O}'$ be the orders and let $f$ be a modular form for which $\operatorname{{\mathscr M}}(\widetilde{\mathscr{O}})^f\neq 0$. Consider the Hecke-linear map

$$\Theta^{\widetilde{\mathscr{O}}}_{\mathscr{O}'}:\operatorname{{\mathscr M}}(\widetilde{\mathscr{O}})\longrightarrow M_{3/2}(4p^2,\varkappa_p).$$

Theta-lift nonvanishing conjecture. If $L(f,1)=0$, then

$$\Theta^{\widetilde{\mathscr{O}}}_{\mathscr{O}'}\bigl(\operatorname{{\mathscr M}}(\widetilde{\mathscr{O}})^f\bigr)=0.$$

If $L(f,1)\neq 0$, then

$$\Theta^{\widetilde{\mathscr{O}}}_{\mathscr{O}'}\bigl(\operatorname{{\mathscr M}}(\widetilde{\mathscr{O}})^f\bigr)\neq 0,$$

unless $f$ and $\mathscr{O}'$ satisfy condition (A) or (B) of Proposition: (A) $f\otimes p^\ast$ has level $p$ and $\epsilon(f,p^\ast)=\left(\frac{-1}{p}\right)\sigma(\mathscr{O}')$, or (B) $f\otimes p^\ast$ has level $p^2$ and $\epsilon(f,p^\ast)=\left(\frac{-1}{p}\right)$. Here $p^\ast=\left(\frac{-1}{p}\right)p$ is the prime discriminant associated to $p$.

This predicts that the theta lift detects the central value $L(f,1)$, apart from the explicitly identified exceptional cases in Proposition 2. The preceding proposition proves vanishing under those exceptional root-number conditions, while the stated implications remain conjectural.