Integer-spectrum conjecture for the n-Queens' graph

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, let $\sigma(\mathcal{Q}(n))$ denote its spectrum, and let $\mathbb{Z}$ denote the integers. Integer-spectrum conjecture. For $n\geq4$,

$$\sigma(\mathcal{Q}(n))\cap\mathbb{Z}=\begin{cases} \left\{-4,n-4\right\},&\text{if $n$ is even;}\\ \left\{-4,-3,\ldots,\frac{n-11}{2}\right\}\cup\left\{\frac{n-5}{2},\ldots,n-5,n-4\right\},&\text{if $n$ is odd.} \end{cases}$$

Furthermore, when $n$ is even, the eigenvalue $n-4$ has multiplicity $(n-2)/2$; when $n$ is odd, the eigenvalue $n-4$ has multiplicity $(n+1)/2$, and the eigenvalues $-3,-2,\dots,\frac{n-11}{2},\frac{n-5}{2},\dots,n-6,n-5$ are simple. The claim is motivated by computations for several values of $n$; the source gives no resolution, so its status remains open.