Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the probability that such a uniformly chosen $n$ by $n$ Toeplitz matrix is singular. Is it known that

$$\lim_{n\to \infty} p_n = 0 ?$$

The equivalent question for random Bernoulli matrices was resolved by Komlós (1963).

I see Probability of random (0,1) Toeplitz matrix being invertible where the exact value of $p_n$ was asked for (and with no answer to the part related to my question).