Nielsen-type conjecture for the mapping class group of a hyperbolic surface

Let $\widehat{\Sigma}$ be the universal hyperbolic surface associated with a compact surface $\Sigma$, let $L$ be the relevant limit set, and fix $x\in L$. The fundamental germ is identified with the inverse-limit-type intersection

$$[\negmedspace [ \pi ]\negmedspace ]_{1} (\widehat{\Sigma},x)\cong\bigcap_{[\pi_{1}\Sigma:G]<\infty}{}^{\ast}G\subset{}^{\ast}\pi_{1}\Sigma.$$

Nielsen-type conjecture. The natural monomorphism

$${\sf MCG}(\widehat{\Sigma},L)\hookrightarrow {\sf Out}([\negmedspace [ \pi ]\negmedspace ]_{1} (\widehat{\Sigma},x))$$

is an isomorphism. Here ${\sf MCG}(\widehat{\Sigma},L)$ denotes the mapping class group of the pair $(\widehat{\Sigma},L)$, and ${\sf Out}$ denotes the outer automorphism group.

This is formulated as a Nielsen-type analogue of the preceding theorem identifying the Teichmüller modular group with the mapping class group. The source explains the injective map through virtual automorphisms and states the surjectivity assertion as conjectural; no resolution is supplied here.