Rationality conjecture for Jacobian points and twisted Eisenstein series

Let $X_0(l)$ be the modular curve with its natural ${\mathbb Q}$-structure and Jacobian, and let $h$ determine the degree-zero twist sheaf ${\mathcal F}_0(h)$ on $X_0(l)$. For each weight $k$, write $E_{k,i;h}$ for the associated twisted Eisenstein series, indexed by $i$, and let $R_k(q)$ be a formal power series.

Rationality conjecture. The sheaf ${\mathcal F}_0(h)$ gives a rational point on the Jacobian if and only if, for each $k$, there exists a nonzero formal power series $R_k(q)$ such that all coefficients of $E_{k,i;h}R_k(q)$ are rational for every $i$. Equivalently, the ratios

$$\frac{E_{k,i;h}}{E_{k,j;h}}$$

have rational Fourier coefficients for all $k$, $i$, and $j$ whenever $E_{k,j;h}\not\equiv 0$.

This conjecture proposes that rationality of the Jacobian point represented by the twisted sheaf is exactly detected by rational Fourier-coefficient data from the associated twisted Eisenstein series. The abstract gives the equivalent formulation in terms of ratios of twisted Eisenstein series, while the supplied text provides no evidence that the conjecture has been resolved.