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on Zhi-Wei Sun's prime-difference matrix determinant conjecture

The statement

For each positive integer $n$ , let $M_n=(m_{ij})_{1\le i,j\le n}$ be the $n\times n$ matrix with integer entries defined by

m_{ij}=\begin{cases}1,&\text{if }|i-j|\text{ is prime},\\0,&\text{otherwise}.\end{cases}

Prove that $\det(M_n)\ne 0$ for every integer $n>3$ , or give an integer $n>3$ for which $\det(M_n)=0$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.