Undecidability of profinite triviality for hyperbolic groups

Let $\Gamma$ be a hyperbolic group, and let $\widehat{\Gamma}$ denote its profinite completion.

Hyperbolic profinite-triviality conjecture. There is no algorithm that can determine whether or not a given hyperbolic group $\Gamma$ has

$$\widehat{\Gamma}\cong 1.$$

The conjecture proposes undecidability of the triviality problem for profinite completions in the hyperbolic case. The source notes that the triviality problem is solvable for hyperbolic groups, and explains that the conjecture would be false if every non-trivial hyperbolic group had non-trivial profinite completion; its status is therefore not settled by the supplied text.