The extended image conjecture for the Dehn–Nielsen–Baer map

Let $S$ be an infinite-type surface properly homotopy equivalent to a graph $\G$, and let $\widetilde{\Theta}:\operatorname{ExMap}(S)\to\operatorname{Map}(\G)$ be the continuous injective homomorphism induced by a proper homotopy equivalence. Let $\mathcal{B}$ be the set of properly immersed oriented curves and lines in $\G$ arising from the components of $\partial S$. Denote by $\operatorname{Map}^{\star}(\G,\mathcal{B})<\operatorname{Map}(\G)$ the subgroup of mapping classes that fix the set of these lines and curves up to homotopy, possibly permuting their classes and reversing orientations. Extended image conjecture. The image of $\widetilde{\Theta}$ is

$$\operatorname{Map}^{\star}(\G,\mathcal{B}).$$

This is the analogue for the extended mapping class group, which allows additional maps such as fractional Dehn twists and therefore permits permutation and orientation reversal of the boundary classes. The conjecture remains open.