Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row by the first $n^2$ elements of the Thue–Morse sequence with indexes from $0$ to $n^2-1$. Let $\mathcal D_n$ be the determinant of this matrix. For example, $$\mathcal D_7=\left| \begin{array}{ccccccc} 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ \end{array} \right|$$ For what $n$ the determinant $\mathcal D_n\ne0$?

I found $\mathcal D_2 = -1,\,\,\mathcal D_{11} = 9,\,\,\mathcal D_{13} = -9,\,\,\mathcal D_{19} = 270,\,\,\mathcal D_{23} = -900,$ and no other cases for $n<1000$. Are there any other cases?