Let $X$ be a smooth projective variety defined over a number field $F$ and consider the Abel-Jacobi map $\mathrm{AJ}_k:\mathrm{CH}_0^k(X_{\overline{\mathbb{Q}}})\rightarrow \mathrm{Jac}^{2k-1}(X)$, where the LHS is the subgroup of the Chow group consisting of cohomologically trivial cycles and the RHS is the intermediate Jacobian considered as complex tori. In general, it is a conjecture that if the AJ image of a cycle is zero, then the cycle is zero, i.e., rationally equivalent to zero.

My question is about a simple situation: suppose $\mathrm{H}^1(X,\mathbb{C})$ vanishes so that $\mathrm{AJ}_1$ is the trivial map, would the whole of $\mathrm{CH}_0^1(X_{\overline{\mathbb{Q}}})$ Or $\mathrm{CH}_0^1(X_{\overline{\mathbb{Q}}})\otimes_{\mathbb{Z}} \mathbb{Q}$ be trivial?