Conjecture on morphisms from punctured-surface homeomorphism groups

Let $S$ be a closed surface, let $D$ be an open disc in $S$, and write $S-D$ for the surface obtained by removing $D$. Let $\mathrm{Homeo}_{0}(S-D)$ be the identity component of the group of homeomorphisms of $S-D$ whose support is contained in the interior of $S-D$. The punctured-surface morphism conjecture. Every group morphism

$$\mathrm{Homeo}_{0}(S-D)\longrightarrow\mathrm{Homeo}_{0}(S)$$

is induced by an inclusion of $S-D$ in $S$. The conjecture concerns the rigidity of morphisms between homeomorphism groups of surfaces and is presented as an attainable problem; no resolution is given in the source.