The conjecture on SDP recovery for k-means at separation greater than $2+\epsilon$

Let the data and cluster model be under the same setting as in Theorem, and let $\Delta$ denote the cluster separation. Consider the SDP relaxation for the $k$-means objective.

SDP recovery conjecture. The SDP relaxation recovers the clusters with high probability whenever

$$\Delta > 2+\epsilon.$$

The preceding theorem establishes recovery under the stronger separation condition $\Delta > 2\sqrt{2}(1+\sqrt{1/m})$, while the paper notes that this bound is not tight and conjectures recovery at separation greater than $2+\epsilon$.