Splitting the degree extension for full mapping class groups

Determine whether the exact sequence 1 → MCG(M) → MCG^±(M) → Z/2 → 1 splits for every simply connected closed 5-manifold M.

Background

The full mapping class group MCG^±(M) contains the orientation-preserving mapping class group MCG(M) as an index-two subgroup, with quotient detected by degree. The paper constructs a splitting for S^2 × S^3, but states that the general splitting problem for simply connected closed 5-manifolds is unresolved.