Nonvanishing conjecture for Gross-curve twists when q is 3 modulo 8

Let $q$ be a prime with $q\equiv3\pmod 8$, let $K$ be the associated imaginary quadratic field, let $H$ be its Hilbert class field, and let $E/H$ be the elliptic curve defined in the source. Its Hasse–Weil $L$-function is denoted by $L(E/H,s)$.

Nonvanishing conjecture. For every prime $q$ satisfying $q\equiv3\pmod 8$,

$$L(E/H,1)\ne0.$$

The conjecture is motivated by numerical calculations and is attributed in the source to Conjecture 1.5 of the cited reference. The authors state that they currently see no way to attack it using Iwasawa theory.