Mazur–Rubin–Stein conjecture on shifted modular-symbol convolutions

Let $q$ be a positive integer, let $\Gamma_0(q)$ be the subgroup of determinant-one integer matrices whose lower-left entry is divisible by $q$, and let

$$f(z)=\sum_{n=1}^{\infty}a(n)e^{2\pi i n z}=\sum_{n=1}^{\infty}A(n)n^{1/2}e^{2\pi i n z}$$

be a newform of weight $2$ for $\Gamma_0(q)$. For $r\in\mathbb Q$, define

$$\langle r\rangle^+=2\pi\int_{i\infty}^r\operatorname{Re}(if(z)\,dz),\qquad \langle r\rangle^-=2\pi i\int_{i\infty}^r\operatorname{Re}(f(z)\,dz),$$

and, for $x\in[0,1]$ and $M\in\mathbb N$, set

$$G_M^{\pm}(x)=\frac{1}{M}\sum_{0\leq a\leq Mx}\left\langle\frac{a}{M}\right\rangle^{\pm}.$$

Mazur–Rubin–Stein conjecture. For each $x\in[0,1]$, the limits satisfy

$$\lim_{M\to\infty}G_M^+(x)=\frac{1}{2\pi}\sum_{n\geq1}\frac{a(n)\sin(2\pi n x)}{n^2},$$

and

$$\lim_{M\to\infty}G_M^-(x)=\frac{1}{2\pi i}\sum_{n\geq1}\frac{a(n)(\cos(2\pi n x)-1)}{n^2}.$$

This conjecture concerns the limiting behaviour of averaged modular symbols associated with a weight-$2$ newform and arose from questions about ranks of elliptic curves over cyclic extensions of $\mathbb Q$. It was formulated by Mazur, Rubin, and Stein based on theoretical and computational evidence; the supplied source does not indicate that it has been resolved.