Existence of overtwisted contact structures on odd-dimensional spheres

Let $S^{2n+1}$ denote the $(2n+1)$-dimensional sphere, and let a contact manifold mean a manifold equipped with a contact structure. Sphere overtwistedness conjecture. The sphere $S^{2n+1}$ admits an overtwisted structure, and consequently any contact manifold admits an exotic overtwisted structure. This would provide overtwisted contact structures on spheres and substantially broaden the supply of exotic overtwisted contact manifolds beyond the constructions established in the paper.