Determine the exact Borel rank of the strong topological Rokhlin property class

Determine the exact Borel rank of the class of countable groups having the strong topological Rokhlin property.

Background

The paper studies the descriptive complexity of the class $\mathsf{STRP}$ of countable groups whose Cantor-action conjugacy action has a comeager orbit, using the standard Borel space of countable groups. It establishes that $\mathsf{STRP}$ belongs to $\boldsymbol{\Pi}^0_4$ and is $\boldsymbol{\Sigma}^0_2$-hard.

The paper also proves that the locus of finitely presented groups with the strong topological Rokhlin property is $\boldsymbol{\Sigma}^0_3$. Consequently, if $\mathsf{STRP}$ is not $\boldsymbol{\Sigma}^0_3$, then some non-finitely-presented group has the property. The exact Borel rank of $\mathsf{STRP}$, however, is left unresolved.