Twisted conjugation model for the 3-spherical quandle

Determine whether the 3-dimensional spherical quandle S^3_ℝ is isomorphic, as a quandle, to a twisted conjugation quandle Conj(H, ψ) for some group H and automorphism ψ ∈ Aut(H).

Background

The quandle S3_ℝ is quandle-isomorphic to Core(SU(2)). It has been shown that for no ψ ∈ Aut(SU(2)) is Conj(SU(2), ψ) isomorphic (as a quandle) to Core(SU(2)). Nevertheless, it is unknown whether there exists some other group H and automorphism ψ for which Conj(H, ψ) is isomorphic to S3_ℝ.

This specific question motivated the broader Problem about realizing core quandles as twisted conjugation quandles.

References

However, we do not even know whether S{3}_{\mathbb{R}} is isomorphic, as a quandle, to any twisted conjugation quandle on a group.

— An embedding of spherical quandles into Lie groups  (2603.29479 - Suzuki et al., 31 Mar 2026) in Section 7 (Results on the 3-spherical quandle S^3_ℝ)