Mock rational point divisor conjecture for rank-2 bielliptic modular curves

Let $X$ be a bielliptic modular curve of genus $2$ and level $N$, and write $Jac(X)$ for its Jacobian. A mock rational point over $\mathbb{Q}(\sqrt{D})$ is a point of the type considered by the quadratic Chabauty computations in the source. Mock rational point divisor conjecture. If $Jac(X)$ has rank $2$ over $\mathbb{Q}$ and $X$ has a mock rational point over $\mathbb{Q}(\sqrt{D})$, then $D$ divides $N$.

This conjecture is based on the explicit computations for genus $2$ modular curves satisfying the paper's LMFDB conditions; the source explicitly says that the resulting list of algebraic points is not provably exhaustive, and gives no resolution of the conjecture.