Orientability conjecture for graph coloring manifolds

Let $G$ be the complement of the $1$-skeleton of a flag simplicial PL sphere, and let $s$ be the maximal valency of $G$. The associated graph coloring manifolds are the Hom complexes

$${\rm Hom}(G,K_n).$$

Orientability conjecture. Every graph coloring manifold is orientable; in particular,

$${\rm Hom}(G,K_n)$$

is orientable also when $\chi(G)\leq n<s+3$. The theorem quoted in the source proves simple connectivity, and hence orientability, for $n\geq s+3$; the conjectural range is the remaining range of chromatic parameters.