Rank-one joint Gaussian distribution conjecture for elliptic-curve twists

Let $E/\mathbb{Q}$ be an elliptic curve given by $y^2=F(x)$ for a monic cubic integral polynomial $F$, and let $K$ be the splitting field of $F$ over $\mathbb{Q}$. Let $G=\operatorname{Gal}(K/\mathbb{Q})$, and for $g\in G$ let $c(g)$ be the integer such that $c(g)-1$ is the number of fixed points of $g$. Define

$$\mu(E)=-\frac{1}{2}-\frac{1}{|G|}\sum_{g\in G}\log c(g),\qquad \sigma(E)=1+\frac{1}{|G|}\sum_{g\in G}(\log c(g))^2.$$

Let $\mathcal{F}$ denote the rank-one quadratic-twist family considered in the source, and let $\Xi_E$ and $\mathfrak{K}_E$ be given by

$$\Xi_E(\underline{\alpha},\underline{\beta})=\int_{(\alpha_1,\beta_1)\times(\alpha_2,\beta_2)}\frac{1}{2\pi\sqrt{\det(\mathfrak{K}_E)}}e^{-\frac12{\bf v}^{\mathrm{T}}\mathfrak{K}_E^{-1}{\bf v}}\,d{\bf v},\qquad \mathfrak{K}_E=\begin{pmatrix}1&\sigma(E)^{-1}\\sigma(E)^{-1}&1\end{pmatrix}.$$

Rank-one joint distribution conjecture. As $d$ ranges over $\mathcal{F}$, the joint distribution of $\log|L'(\frac12,E_d)|$ and $\log(|\Sha(E_d)|R(E_d)/\sqrt{|d|})$ is asymptotically bivariate normal in the precise sense that, for fixed $\underline{\alpha}=(\alpha_1,\alpha_2)$ and $\underline{\beta}=(\beta_1,\beta_2)$, the count displayed in the source statement is asymptotic to

$$(\Xi_E(\underline{\alpha},\underline{\beta})+o(1))\#\{d\in\mathcal{F}:20<|d|\le X\}.$$

Consequently, the second coordinate has approximately Gaussian mean $(\mu(E)+1)\log\log|d|$ and variance $\sigma(E)^2\log\log|d|$. This is the proposed rank-one analogue of the conjecture for the rank-zero family and is motivated by BSD and the preceding lower-bound theorem; it remains unproved in the stated generality.