Injectivity of the dual winding quotient map for strong Weil curves of odd analytic rank

Let $E$ be a strong Weil curve over $\mathbb{Q}$ of conductor $M\mid N$. Suppose that $E$ has odd analytic rank, and let $\xi_{E,N}^\vee$ be the dual map associated with the winding quotient construction. Injectivity conjecture. The map $\xi_{E,N}^\vee$ is injective. The preceding result only establishes injectivity up to a possible $2$-group in the kernel; computational evidence suggests that this $2$-group cannot occur, but a general proof is not known.