Computably simple copies for homogeneous completely decomposable groups

Let $H$ be a subgroup of $\mathbb{Q}$, and consider a computable group of the form

$$\bigoplus_{i\in\omega} H.$$

A computably simple-copy conjecture. Theorem~ holds for any computable group of this form. That is, every computable homogeneous completely decomposable group of infinite rank has a computably simple copy. The claim extends the construction for the free abelian group; the nonhomogeneous or non-completely-decomposable cases are described as substantially more difficult because divisibility conditions must be maintained.