Vu's distinct singular values conjecture for symmetric Rademacher matrices

Let $M_n^{sym}$ be a symmetric random matrix whose entries are independent Rademacher variables. Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singular values of $M_n^{sym}$ are distinct. This question was posed by Vu in the cited survey and remains open to the best of the authors' knowledge.