Polynomial-factor concentration conjecture for determinants of random matrices

Let $c \le C$ be positive constants. Let $A$ be the random matrix of size $n$ whose entries are independent random variables with mean zero and variances between $c$ and $C$. Polynomial-factor concentration conjecture. With probability tending to one,

$$|{\rm {\det}} A| = n^{O(1)}\mathop{\mathbf{E}} |{\rm {\det}} A|, \qquad {\rm {\det}} A^2 = n^{O(1)}\mathop{\mathbf{E}}({\rm {\det}} A^2).$$

The conjecture would strengthen the paper's sub-exponential-factor estimate for determinant-based permanent estimators to a polynomial-factor approximation. It is motivated by the corresponding Gaussian result, while the analogous general central-limit statement attributed to Girko is described as believed but unproved by the authors.