Vanishing conjecture for the arithmetic theta lift

Let $f\in H_{2-n}(\omega_\mathscr{L})^\Delta$, and let

$$\xi:H_{2-n}(\omega_\mathscr{L})\longrightarrow S_n(\overline{\omega}_\mathscr{L})$$

be the complex-conjugate-linear differential operator. Let $\widehat{\theta}_\mathscr{L}(f)$ be the arithmetic theta lift.

Vanishing conjecture. If $\xi(f)=0$, then

$$\widehat{\theta}_\mathscr{L}(f)=0.$$

This is the kernel-vanishing formulation of the assertion that the arithmetic theta lift factors through $\xi$. Its status is not resolved in the supplied source.