Conjecture on spheres with unique smooth structures

For an integer $n>4$, let $S^n$ denote the $n$-dimensional sphere, and say that it has a unique smooth structure when every smooth manifold homeomorphic to $S^n$ is diffeomorphic to it. Unique smooth structure conjecture. For dimensions greater than $4$, the only spheres with a unique smooth structure are $S^5$, $S^6$, $S^{12}$, $S^{56}$, and $S^{61}$. This conjecture extends the dimensions currently known to have unique smooth structures; the paper establishes the claim through dimension $61$, while work of Behrens, Hill, Hopkins, and Mahowald indicates that the next such sphere, if it exists, has dimension at least $126$.