Normal-distribution and variance conjecture for modular symbols

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$ with squarefree conductor $N$, and define

$$\mathcal{C}_E=\frac{6}{\pi^2}\prod_{\ell\mid N}(1+\ell^{-1})^{-1}L(\operatorname{Sym}^2(E),1).$$

For $m\geq 1$, define

$$\operatorname{Var}(m)=\frac{1}{\varphi(m)}\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times}([a/m]^+)^2.$$

Modular-symbol distribution conjecture. As $m$ tends to infinity, the distribution of the values $\{[a/m]^+/\sqrt{\log(m)}:a\in(\mathbb{Z}/m\mathbb{Z})^\times\}$ converges to a normal distribution with mean zero and variance $\mathcal{C}_E$. Moreover, for every divisor $\kappa$ of $N$, there is a constant $\mathcal{D}_{E,\kappa}\in\mathbb{R}$ such that

$$\lim_{\substack{m\to\infty\\(m,N)=\kappa}}\bigl(\operatorname{Var}(m)-\mathcal{C}_E\log(m)\bigr)=\mathcal{D}_{E,\kappa}.$$

Petridis and Risager proved an averaged distribution theorem over $m\leq X$; this conjecture concerns each modulus tending to infinity and includes a refined second-order variance statement.