Nonvanishing of specializations of the algebraic anticyclotomic $p$-adic $L$-function

Let $\mathbf f$ be the Hida family over $K$, let $L_p^{\rm alg}(\mathbf f/K)\in\mathbb I[[\Gamma_\infty]]$ be its algebraic anticyclotomic $p$-adic $L$-function, and let $\kappa$ denote an arithmetic specialization. Let $w_{\mathbf f}\in\{\pm1\}$ be the generic root number. Nonvanishing conjecture. If $w_{\mathbf f}=1$, then

$$\kappa\bigl(L_p^{\rm alg}(\mathbf f/K)\bigr)\neq 0.$$

This is an application to a further conjecture from the cited work. The supplied text does not state whether this assertion is proved in the paper or remains open.