Optimal condition-number threshold conjecture for low-rank synchronization factorization

Let $\boldsymbol{L}\in\mathbb{R}^{nd\times nd}$ be positive semidefinite and let $\boldsymbol{Z}$ denote the synchronization ground-truth matrix, with

$$\boldsymbol{L}\boldsymbol{Z}=0.$$

Optimal condition-number threshold conjecture. The optimization landscape of the low-rank factorization problem is benign if

$$\frac{\lambda_{\max}(\boldsymbol{L})}{\lambda_{d+1}(\boldsymbol{L})}<\frac{2p}{d+1}.$$

Here, benign means that the factorized problem has no spurious second-order critical points. The conjectured threshold matches the twisted-state counterexample threshold and would improve the current state-of-the-art global landscape guarantee, whose bound differs from $2p/(d+1)$ by roughly a factor of four.