Existence of smooth Sasaki–Einstein manifolds from iterated $S^3$ joins

An iterated $S^3$ join in the Gorenstein case is a smooth manifold of odd dimension. Existence conjecture. In the Gorenstein case, the iterated $S^3$ joins admit smooth Sasaki–Einstein manifolds in all odd dimensions. This proposes that the iterative construction can be continued indefinitely without taking products; the source indicates that an explicit proof is not currently available.