Sasakian–Einstein metrics on odd-dimensional homotopy spheres bounding parallelizable manifolds

Let $M$ be an odd-dimensional homotopy sphere that bounds a parallelizable manifold. Sasakian–Einstein metric conjecture. Every such $M$ admits a Sasakian–Einstein metric. This would extend the known result for standard and Kervaire spheres in dimensions congruent to $1$ modulo $4$; the claim is motivated by the difficulty of treating the remaining elements of the Kervaire–Milnor groups in dimensions congruent to $3$ modulo $4$.