Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two positions above the main antidiagonal, and with zeros everywhere else. For example, we have that:
$$M_{7} =\left( \begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}\right).$$
I refer to matrices of this form as Arndt matrices, based on the following conjecture due to Joerg Arndt (see https://oeis.org/A047211) which has been tested up to $n=177$:
Conjecture (Arndt, 2011): The characteristic polynomial of $M_{n}$ is irreducible over $\mathbb{Q}$ if and only if $n$ is congruent to an element in $\{ 2, 4 \}$ modulo $5$.
I have been interested in this conjecture for some time, and have made numerous attempts to prove this conjecture.
My first attempt at proving this conjecture was to try to find a general technique for row-reducing matrices of the form $x I_{n} - M_{n}$ in order to evaluate $\text{det}(x I_{n} - M_{n})$. However, the process of row-reducing matrices of this form is very complicated. I have also considered using the Leibniz formula for determinants and cofactor expansion to evaluate $\text{det}(x I_{n} - M_{n})$, but this also seems to be very complicated. I have also considered using a recursive/inductive approach, by considering the possibility of expressing $\text{det}(x I_{n} - M_{n})$ in terms of expressions of the form $\text{det}(x I_{m} - M_{m})$ for $m<n$, but it is not clear how to construct such a recursive formula.
I have several questions related to Arndt's conjecture, listed below:
(1) Is there a simple way of evaluating $\text{det}(x I_{n} - M_{n})$? Is there a simple combinatorial formula for the coefficients of $\text{det}(x I_{n} - M_{n})$?
(2) Is there an intuitive/heuristic explanation as to "why" the above conjecture may be true?
(3) Do you have any suggestions or general insights as to how to approach the problem of proving the above conjecture?