Non-vanishing conjecture for quadratic twists of the Gross curve

Let $q$ be a prime with $q \equiv 3 \ \mathrm{mod}\ 8$, let $K=\mathbb{Q}(\sqrt{-q})$, let $H$ be the Hilbert class field of $K$, and let $A$ be the Gross elliptic curve over $H$. Write $A^{(-q)}$ for its quadratic twist by $-q$. Non-vanishing conjecture. For all primes $q$ with $q \equiv 3 \ \mathrm{mod}\ 8$, we have

$$L(A^{(-q)}/H, 1) \neq 0.$$

The paper presents this as a conjectural analogue of the preceding non-vanishing theorem for primes congruent to $7$ modulo $8$. Its status is not resolved in the supplied text.