The sharp lower bound conjecture for edge eigenvector Gaussian approximation

Let $\mathcal{G}_{d,N}$ denote the space of random $d$-regular graphs on $N$ vertices, and let $T_N: \mathcal{G}_{d,N} \to \mathbb{R}$ be a sequence of statistics that distinguishes edge eigenvector distributions from the standard Gaussian distribution $\mathcal{N}(0,1)$. Sharp lower bound conjecture. There exists a constant $c>0$ such that

$$\liminf_{N \to \infty} N^{1/6} \cdot \operatorname{dist}_{\mathrm{TV}}(T_N,\mathcal{N}(0,1)) \geq c.$$

The conjecture asserts that the $N^{-1/6}$ convergence scale suggested by the local-resampling, moment, and constrained-DBM approaches is optimal for edge eigenvector universality. The supplied discussion presents this as strongly supported by several methods, but does not establish the lower bound or specify a proof for every distinguishing sequence of statistics.