Smooth unknottedness of turned ±1-twist spun tori

Ascertain whether the turned ±1-twist spun torus σ_T^{±1}(K) of a classical knot K is smoothly unknotted in S⁴.

Background

Juhász and Powell proved that σ_T^{±1}(K) is topologically unknotted, answering a question of Boyle in the topological category. The authors pose the smooth analogue: whether σ_T^{±1}(K) is smoothly unknotted.

Although they do not resolve this question, they show that the double branched covers Σ₂(σ_T^{2m+1}(K)) are diffeomorphic to S²×S², and in particular Σ₂(σ_T^{±1}(K))≅S²×S², providing evidence and related structural results without settling smooth unknottedness.