Asymptotic maximal determinant conjecture for Bernoulli matrices

Let $M_n$ be an $n\times n$ matrix whose entries are independent random variables taking values $1$ and $-1$ with probability $1/2$. Hadamard's inequality gives

$$|\det M_n|\le n^{n/2}.$$

Maximal determinant conjecture. Almost surely,

$$|\det M_n|=n^{(1/2-o(1))n}.$$

The claim asks whether a random Bernoulli matrix has determinant exponentially close, on the logarithmic scale, to the Hadamard upper bound. The source presents this as a formerly common conjecture and does not give a resolution.