Proper 3-realizability conjecture for finitely generated one-relator groups

Let $G$ be a finitely generated one-relator group. Theorem 1.1 establishes proper $3$-realizability under the hypothesis that $G$ has finitely many ends. Proper 3-realizability conjecture. The “finitely many ended" hypothesis in Theorem 1.1 can be omitted. Thus every finitely generated one-relator group should be properly $3$-realizable. This would extend the theorem from groups with finitely many ends to all finitely generated one-relator groups.