Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$.

My question. For any fixed $i\in\{1,\dots,n\}$, is there any known estimate or bound on $$ f(\lambda_1,\dots,\lambda_n):=\prod_{j=1, j\ne i}^n|\lambda_i-\lambda_j| \ \ \ \ \ ? $$ In particular, I wonder whether $f(\lambda_1,\dots,\lambda_n)^{1/n}$ converges a.s. to a nonzero constant for $n\to\infty$ (as suggested by numerical experiments).

$^{1}$i.e., a random Hermitian matrix whose strictly upper triangular entries are i.i.d. sub-gaussian r.v.'s with mean 0 and variance 1 and whose diagonal entries are independent sub-gaussian r.v.'s with mean 0 and variances bounded by $n^{1−o(1)}$, with the diagonal entries independent of the strictly upper triangular entries.