Completely Decomposable Quotient Divisible Abelian Groups with $\mathrm{UA}$-Rings of Endomorphisms

A ring $K$ is called a unique addition ring (a $\mathrm{UA}$-ring) if there exists a unique binary operation $+$ on the multiplicative semigroup $(K,\,\cdot\,)$ of $K$ such that $(K,\,\cdot\,,+)$ is a ring. We say that an abelian group is an $\operatorname{End}$-$\mathrm{UA}$-group if its endomorphism ring is a $\mathrm{UA}$-ring. We find $\operatorname{End}$-$\mathrm{UA}$-groups in the class of completely decomposable quotient divisible abelian groups.