Independence of the comprehension schema in second-order arithmetic from the parameter-free countable choice

We prove that the full Comprehension schema $\mathbf{CA}$ in second-order arithmetic $\mathbf{PA}_2$ is not provable in the subtheory $\mathbf{PA}_2^\ast$ with the parameter-free Comprehension even when adding the parameter-free Countable Choice $\mathbf{AC}_\omega^\ast$ and the Comprehension $\mathbf{CA}(\mathbf\Sigma^1_2)$ for all $\mathbf\Sigma^1_2$ formulas with parameters.