Nonvanishing of quadratic twists at the central value

Let $K$ be a local field of characteristic zero and let $L_K$ be its Weil group when $K$ is archimedean and its Weil–Deligne group when $K$ is non-archimedean. Let $\tau_0=\bigotimes\tau_{0,v}$ be an irreducible unitary cuspidal automorphic representation of $GL_{2n}(\mathbb{A})$ of symplectic type. Say that $\tau_0$ is good at a place $v$ if the local Langlands parameter of $\tau_{0,v}$ has an irreducible symplectic subrepresentation. Quadratic-twist nonvanishing conjecture. If $\tau_{0,v_0}$ is good at some place $v_0$, then there exists a quadratic character $\eta:\mathbb{A}^{\times}/F^{\times}\to\mathbb{C}^{\times}$ such that

$$L\left(\frac{1}{2},\tau_0\otimes\eta\right)\ne0.$$

This is proposed as a weak generalization of a theorem on central $L$-values and predicts nonvanishing after a suitable quadratic twist; its status is not resolved in the supplied text.