The Sasaki–Einstein existence conjecture for Smale 5-manifolds

A Smale $5$-manifold is a closed simply connected spin $5$-manifold. Let $M_\infty$ denote $S^2\times S^3$, and let $M_2$ be the unique closed simply connected spin $5$-manifold with

$$\mathrm{H}_2(M_2,\mathbb{Z})=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}.$$

For a non-negative integer $k$, write $kM_\infty$ for the $k$-fold connected sum of $S^2\times S^3$.

Sasaki–Einstein existence conjecture. For each integer $k\leq 8$ and $n\geq 2$, the Smale $5$-manifold

$$kM_\infty\#nM_2$$

admits a Sasaki–Einstein metric.

This is one of three conjectures proposed toward the classification of closed simply connected Sasaki–Einstein $5$-manifolds. It asserts existence for the specified family; the supplied source does not state whether the conjecture has been resolved.