I'm reading Ghrist's paper "Configuration spaces and braid groups on graphs in robotics". In this discussion, counterexamples are shown for both Theorem 2.3 and the implication "Theorem 2.3 $\Rightarrow$ Corollary 2.4". Are there other proofs, in literature, that configuration spaces of trees are Eilenberg-MacLane spaces? Or, is there a way to "fix" Ghrist's proof?
contains a proof that for any connected, finite graph having essential vertices (of degree $\ge3$) the "deleted product" $$C_2(\Gamma) = \Gamma\times\Gamma \setminus \Delta\Gamma$$ is aspherical. The proof uses the following Theorem of J.H.C. Whitehead:
Suppose $X=A\cup B$, where $X$, $A$ and $B$ are connected polyhedra. Suppose that $A\cap B$ has finitely many path components $C_i$. If:
then $X$ is aspherical.
I don't know if this proof can be adapted to give asphericity of $C_N(\Gamma)$ for $N>2$ when $\Gamma$ is a tree, but perhaps this is what Ghrist had in mind.