The anticyclotomic Iwasawa nonvanishing conjecture for definite Hida families

Let $f_\mathfrak p^\dagger$ be a non-exceptional arithmetic specialization of weight $k_\mathfrak p\ge2$, let $\chi$ be a finite-order character of $G_\infty$, and let $\chi_\mathfrak p:\mathcal R_\infty\to\overline{\mathbb Q}_p$ be the induced specialization map. Let $\mathcal L_p(f_\infty/K)\in\mathcal R_\infty$ be the two-variable $p$-adic $L$-function. The anticyclotomic Iwasawa nonvanishing conjecture. Assume that $w=1$. Then

$$L_K\bigl(f_\mathfrak p^\dagger,\chi,k_\mathfrak p/2\bigr)\ne0 \quad\Longleftrightarrow\quad \chi_\mathfrak p\bigl(\mathcal L_p(f_\infty/K)\bigr)\ne0.$$

This is the anticyclotomic Iwasawa-theoretic analogue of the preceding nonvanishing conjecture; the source gives no general proof.