On endomorphism rings of $B_1$-groups that are not $B_2$-groups

Finite rank Butler groups are pure subgroups of completely decomposable groups of finite rank and were defined by M.C.R. Butler. Extending this concept to infinite rank groups, Bican and Salce gave various possible descriptions: A $B_2$-group $G$ is a union of an ascending chain of pure subgroups $G_{\alpha}$ such that for every $\alpha$ we have $G_{\alpha+1}=G_{\alpha}+H_{\alpha}$ for some finite rank Butler group $H_{\alpha}$. A $B_1$-group is a torsion-free group $G$ satisfying $\mathrm{Bext}_{\mathbb{Z}}^1(G,T)=0$ for all torsion groups $T$. While the class of $B_2$-groups is contained in the class of $B_1$-groups, it is in general undecidable in ZFC if the two classes coincide. In this paper we study the endomorphism rings of $B_1$-groups which are not $B_2$-groups working in a model of ZFC that satisfies $2^{\aleph_0}=\aleph_4$.