Finite groups as the only abstractly recognizable groups

Let $(\mathcal{G}_{WP},d,\nu_{WP})$ denote the space of marked groups with solvable word problem, equipped with its metric and word-problem representation. An abstract group is abstractly recognizable when its isomorphism class can be recognized from this representation, independently of the marking.

Finite-recognizability conjecture. Finite groups are the only abstractly recognizable groups in $(\mathcal{G}_{WP},d,\nu_{WP})$.

The conjecture concerns recognition of isomorphism classes rather than individual marked groups. The paper presents it as an unanswered question after discussing limits, preformation, and the difficulty of abstract recognition.