Vu's distinct singular values conjecture for random symmetric Bernoulli matrices

Let $M_n^{sym}$ be a random symmetric matrix with Bernoulli $pm 1$ entries.

Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singular values of $M_n^{sym}$ are distinct.

This conjecture is motivated by the study of repeated singular values of random symmetric matrices. The paper proves a strong quantitative version for singular values in a bulk portion of the spectrum, but the full statement remains unresolved.