Infinitely many families of Sasaki–Einstein metrics on odd-dimensional spheres

Let $S^{2n+1}$ denote the $(2n+1)$-dimensional sphere, with $n$ a positive integer. Sasaki–Einstein metric abundance conjecture. There are infinitely many families of Sasaki–Einstein metrics on $S^{2n+1}$ for all $n$. This would extend known constructions on $S^5$ and on connected sums of copies of $S^2\times S^3$, and would provide higher-dimensional existence results for Sasaki–Einstein metrics on odd-dimensional spheres. The statement is presented as an expectation, and no resolution is given here.