The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to a wedge of $n$ circles with all vertices of valence $\geq 3$. A metric graph can be thought of as simply a graph with an assignment of positive edge lengths to all edges, and when an edge length goes to $0$, the edge itself also contracts, yielding a new graph type in $MG_n$. One also often assumes that the sum of edge lengths is $1$, since this condition yields a homotopy equivalent space.

One of the useful properties of $MG_n$ is that it is a rational $K(\pi,1)$ for $\mathrm{Out}(F_n)$: $H_*(MG_n;\mathbb Q)\cong H_*(\mathrm{Out}(F_n);\mathbb{Q})$. If $MG_n$ were an actual $K(\pi,1)$, the fundamental group would be $\mathrm{Out}(F_n)$ and that would be that, but it seems in practice that $MG_n$ is fairly highly connected, and in particular simply connected. $MG_2$ is certainly contractible. I'm not sure about $MG_3$; I would guess it's contractible. I suspect that $MG_4\simeq S^4$.

My question is whether $MG_n$ is indeed simply connected. I am unaware of any particular consequences this would have, but it also seems we ought to know such a basic property about such a fundamental space.