Galatius–Randal-Williams' conjecture on the class of the homotopy sphere Sigma_Q

Let $n\geq 3$ be odd, and let $\Sigma_Q$ be the homotopy sphere obtained as the boundary of the manifold $Q$ in the relevant bordism group. Its class $[\Sigma_Q]$ lies in $\operatorname{coker}(J)_{2n+1}$. Galatius–Randal-Williams' conjecture.

$$[\Sigma_Q]=0.$$

This conjecture asserts that the class of $\Sigma_Q$ vanishes for every odd $n\geq 3$. The statement was previously known for $n=3,7$ and for $n\equiv 1\pmod 4$; the general case is left open in the source.