Bordenave–Caputo–Chafaï–Tikhomirov conjecture on the spectral radius without a fourth moment

Let $X_N=(X_{i,j})_{i,j=1}^N$ be a random $N\times N$ matrix whose entries are independent copies of a complex-valued random variable $\mathbf{x}$ satisfying

$$\mathbb{E}[\mathbf{x}]=0\quad\text{and}\quad\mathbb{E}[|\mathbf{x}|^2]=1.$$

Let $\rho(X_N)$ denote the spectral radius of $X_N$:

$$\rho(X_N):=\max\{|\lambda|:\text{$\lambda$ is an eigenvalue of $X_N$}\}.$$

Bordenave–Caputo–Chafaï–Tikhomirov conjecture. The convergence in probability

$$\lim_{N\to\infty}\frac{\rho(X_N)}{\sqrt{N}}=1$$

holds under the sole assumptions $\mathbb{E}[\mathbf{x}]=0$ and $\mathbb{E}[|\mathbf{x}|^2]=1$. The circular law gives the corresponding lower bound, while the upper bound was known under a finite fourth-moment assumption; the conjecture asserts that no fourth moment is needed, equivalently that there are no outliers in the circular law.