Main conjecture on computable functions on the space of marked groups

Let $\mathcal{G}_{WP}$ be the space of marked groups with solvable word problem, equipped with its computability structure, and let $\Lambda_{WP}$ denote the corresponding representation. A function is $\Lambda_{WP}$-computable when it is computable relative to this representation; a recursive metric space is a metric space with effective computable structure.

Main conjecture. Any $\Lambda_{WP}$-computable function defined on $\mathcal{G}_{WP}$, with image in any recursive metric space, is continuous.

The paper explains that standard effective continuity theorems do not apply to $\mathcal{G}_{WP}$, while known examples of computable discontinuous functions on recursive metric spaces are pathological. Whether every computable function in this setting is continuous remains open.