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 (over $\mathbb{R}$). 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 http://mathoverflow.net/questions/128961/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). 


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For $n = 1,\dots,11$ the number of singular matrices is $1,4,13,50,153,522,1637,5065,14337,39755,104088$.  Sadly this is not in OEIS.