Agashe's divisibility conjecture for quadratic twists of optimal elliptic curves

Let $E/\mathbb{Q}$ be an optimal elliptic curve of conductor $N$, let $-D$ be a negative fundamental discriminant such that $D$ is coprime to $N$, and let $E^{-D}/\mathbb{Q}$ denote the twist of $E/\mathbb{Q}$ by $-D$. Suppose that $L(E^{-D},1)\neq 0$.

Agashe's conjecture. Up to a power of $2$, the square of the order of $E^{-D}(\mathbb{Q})$ divides

$$|\Sha(E^{-D}/\mathbb{Q})|\cdot\prod_{p\mid N}c_p(E^{-D}).$$

This conjecture is motivated by the rank-zero Birch–Swinnerton-Dyer formula, which identifies the corresponding quotient involving the period with the Shafarevich–Tate group, Tamagawa numbers, and the square of the rational-point group. The source paper proves a slightly more general statement without the optimality hypothesis, so the conjecture is solved in the setting considered here.