Isomorphism conjecture for torus-link and circular-group presentations

Let $n,m\in\mathbb{N}^*$ and let $d$ be the index set used for the generators $m_i$ and $a_i$ in the torus-link and circular-group presentations. Consider the group with Presentation

$$and the group with Presentation$$

. Define a morphism by

$$x\longmapsto a_1\cdots a_m,\qquad y\longmapsto a_1\cdots a_n,\qquad m_i\longmapsto a_i\quad(i\in[d]).$$

Torus-link/circular-group isomorphism conjecture. For all $n,m\in\mathbb{N}^*$, this morphism is an isomorphism. The claim generalises the previously established coprime case; the supplied text notes that an inverse is not known from the presentations, although the torus-link presentation is already known there to define a group isomorphic to $G(n,m)$.