Hopficity conjecture for labeled Thompson groups

Let $G$ be a finitely generated group. The labeled Thompson group $V(G)$ is defined in the paper's preceding discussion.

Hopficity conjecture. $V(G)$ is Hopfian if and only if $G$ is Hopfian.

This conjecture concerns whether the labeled Thompson group construction preserves and reflects the property that every surjective endomorphism is an isomorphism. Its status is not specified in the source material provided.