Conjecture on the largest Fiedler-vector entry in the stochastic block model

Let the stochastic block model have $n$ vertices, let $\mathbf{v_2}$ denote its Fiedler vector, and let $\mathbf{g}$ be the corresponding indicator vector. The infinity norm is

$$\left\| \mathbf{v_2} - \frac{\mathbf{g}}{\sqrt{n}} \right\|_{\infty}.$$

Largest-entry conjecture. The largest entry of $\mathbf{v_2}$ satisfies

$$\left\| \mathbf{v_2} - \frac{\mathbf{g}}{\sqrt{n}} \right\|_{\infty} \sim \frac{\log n}{n}.$$

This conjecture concerns the extremal behavior of the Fiedler vector and quantifies the deviation of its most extreme entries from the normalized community-indicator vector. The surrounding discussion motivates it using the higher classification accuracy observed for vertices with extreme-magnitude Fiedler-vector entries; its resolution is not specified in the source.