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on Determinants of the Fibonacci-sum matrix

The statement

For each integer $n\ge 1$ , define the integer matrix $M_n=(m_{ij})_{1\le i,j\le n}$ by

m_{ij}=\begin{cases} 1, & i+j \text{ is a Fibonacci number}, \\ 0, & \text{otherwise}. \end{cases}

Prove that $\det(M_n)\in\{-1,0,1\}$ for every integer $n\ge 1$ .

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.