The conjecture on global synchrony with negative edges

Let $A$ be an $n\times n$ random matrix with zero diagonal and independent off-diagonal entries satisfying

$$A_{ij}=\begin{cases}1&\text{with probability }\frac12+\delta,\\\\-1&\text{with probability }\frac12-\delta.\end{cases}$$

A matrix is globally synchronizing when the only local minima of its Kuramoto energy are the global synchronized minima. Negative-edge synchrony conjecture. For every $\varepsilon>0$, if $\delta\geq(1+\varepsilon)\sqrt{\frac{\log n}{2n}}$, then $A$ is globally synchronizing with high probability. This is motivated by the Burer–Monteiro approach to community detection; high-rank versions are known, but this rank-two-type threshold remains open.