The image conjecture for the Dehn–Nielsen–Baer map of a graph-like surface

Let $S$ be an infinite-type surface with compact boundary, properly homotopy equivalent to a locally finite infinite graph $\G$, and let $\Theta$ be the induced homomorphism from $\operatorname{Map}(S)$ to $\operatorname{Map}(\G)$. Fix a proper homotopy equivalence $\phi:S\to\G$. Let $\mathcal{B}$ be the set of properly immersed oriented curves $S^1\to\G$ and immersed oriented lines $\mathbb{R}\to\G$ that are images under $\phi$ of the components of $\partial S$. Denote by $\operatorname{Map}(\G,\mathcal{B})<\operatorname{Map}(\G)$ the subgroup of mapping classes that fix each line or curve in $\mathcal{B}$, together with its orientation, up to homotopy. Image conjecture. The image of $\Theta$ is

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

This conjecture gives a geometric characterization of the image of the Dehn–Nielsen–Baer homomorphism in terms of the properly immersed curves and lines arising from the boundary of $S$.