Conjecture on finitely many vanishing quadratic twists for higher-weight modular forms

Let $f$ be a newform of level $\Gamma_0(N)$ and even weight $k$ with $k\geq 6$, let $\pi$ be the cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A})$ generated by $f$, and let $\varepsilon(\frac{1}{2},\pi\otimes\eta_E)$ denote the root number associated with the twist by the quadratic character $\eta_E$ of a quadratic field $E$. Finiteness conjecture. There are at most finitely many quadratic fields $E$ such that

$$\varepsilon\left(\frac{1}{2},\pi\otimes\eta_E\right)=1 \quad\text{and}\quad L\left(\frac{1}{2},\pi\otimes\eta_E\right)=0.$$

The assertion strengthens the expected scarcity of central-value vanishings for even-weight modular forms of weight at least $6$. It is explicitly described as still open, and the source notes that not even examples of modular forms satisfying it are known.