Erratum to "Linear representations of 3-manifold groups over rings"

The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form $\mathrm{PGL}_2(R),$ where $R$ is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type $\mathrm{SL}_2, \mathrm{GL}_2, \mathrm{PSL}_2$, or $\mathrm{PGL}_2$ as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for $\mathrm{SL}_2$ and $\mathrm{GL}_2$. Here, we provide a modified argument for the $\mathrm{PSL}_2$ and $\mathrm{PGL}_2$ cases.