Splitting of the SKK short exact sequence in odd dimensions for all twice-stabilised tangential structures

Prove that for every twice stabilised tangential structure ξ and every odd dimension n, the short exact sequence 0 → ⟨S_b^n⟩_{SKK_n} → SKK_n^{ξ} → Ω_n^{ξ} → 0 splits.

Background

The paper establishes a short exact sequence relating SKK-groups to bordism groups, with kernel generated by the bounding sphere. In many odd-dimensional cases the authors construct splittings using the Kervaire semi-characteristic and other tools. Motivated by their systematic positive results across numerous tangential structures, they propose a general conjecture asserting that the short exact sequence should always split in odd dimensions once the structure is twice stabilised.