Existence of a Hopfian finitely presented group with undecidable weak Whitehead problem

A Hopfian finitely presented group is a finitely presented group for which every surjective endomorphism is an automorphism; its word problem asks whether a given word represents the identity, and its weak Whitehead problem asks whether two finite tuples are related by an automorphism. Existence conjecture. There exists a Hopfian finitely presented group with decidable word problem (so a computable group) but undecidable weak Whitehead problem. The conjecture would separate decidability of the word problem from decidability of the weak Whitehead problem for Hopfian finitely presented groups. Combined with the paper's characterization, such a group would provide a computable finitely presented structure without a computable $d$-$\Sigma_2$ Scott sentence.