The rational cuspidal groups conjecture for modular Jacobians

Let $N$ be a positive integer. Let $C_N(\mathbb{Q})$ be the subgroup of $J_0(N)(\mathbb{Q})$ represented by degree-zero cuspidal divisors, and let $C(N)$ be the subgroup formed by classes of $\mathbb{Q}$-rational divisors in $C_N(\mathbb{Q})$, equivalently the divisor classes fixed by the absolute Galois group of $\mathbb{Q}$. Rational cuspidal groups conjecture.

$$C_N(\mathbb{Q})=C(N).$$

The conjecture concerns the first inclusion $C(N)\subseteq C_N(\mathbb{Q})$. It is motivated by computations in which the two groups agree, but remains open in general.