Even-dimensional Pin+ case at n ≡ 2 (mod 8)

Determine whether the short exact sequence 0 → Z → SKK_n^{Pin^+} → Ω_n^{Pin^+} → 0 splits for all even dimensions n ≡ 2 (mod 8) beyond the cases n = 2 and n = 10.

Background

In even dimensions, the authors give complete splitting results for most congruence classes for Pin^+ structures: the sequence does not split for n ≡ 0,4 (mod 8), and it splits for n ≡ 6 (mod 8) and for n = 2,10. The remaining general case n ≡ 2 (mod 8) is left open, with only low-dimensional instances known.