Circular-law universality for constant linear margins

Under the setting of Corollary, let $X$ be the margin-conditioned random matrix and let $\widetilde{X}_{n}$ be its centered and normalized version. The empirical eigenvalue distribution is the probability measure assigning mass $1/n$ to each eigenvalue of $\widetilde{X}_{n}$.

Circular-law universality conjecture. For constant linear margins and every base measure $\mu$ satisfying the hypothesis of Corollary, the empirical eigenvalue distribution of

$$\widetilde{X}_{n}=(2\psi”(\phi(a))n)^{-1/2}(X-a\mathbf{1}\mathbf{1}^{\top})$$

converges weakly in probability to the circular law.

The conjecture extends the circular-law result for uniformly random doubly stochastic matrices to arbitrary base measures satisfying the stated hypotheses. The cited doubly stochastic special case is described as already established by Nguyen, while the universal extension remains conjectural.