Galois-module structure conjecture for quadratic twists

Let $E/\mathbb Q$ be an elliptic curve of rank $1$, let $\mathcal F_b$ be the twist family specified by the local data $b$, and write $\mathcal F_b(X)=\{d\in\mathcal F_b:|d|\leq X\}$. Let $n_b$ be the dimension of the systematic Selmer subspace, and assume the paper's stated elliptic-curve Selmer-rank assumption. Galois-module structure conjecture. One has

$$\lim_{X\rightarrow\infty}\frac{\#\{d\in\mathcal F_b(X):E(\mathbb Q(\sqrt d))/E(\mathbb Q(\sqrt d))_{\mathrm{tors}}\cong\mathbb Z[G]\}}{\#\mathcal F_b(X)}= \begin{cases} 1/2,&n_b=1,\\ 1/8,&n_b=2,\\ 5/64,&n_b=3,\\ 29/1024,&n_b=4. \end{cases}$$

This predicts the frequency of the regular representation as the Mordell–Weil Galois module in the quadratic twist family; it is derived from the preceding frequency conjecture and remains open in the source.