If all the diagonal blocks are not of the same order then all the eigenvalues are not an integers.

Question

Let $M=\begin{pmatrix} \begin{array}{cccccccc} 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\ 1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\ 1 & 1 & 1 & 1 & 1 & 0 & 0 &1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 &0\\ \end{array} \end{pmatrix}\implies\begin{pmatrix} \begin{array}{ccc} 0_{2\times2} & 1_{2\times3} & 1_{2\times2}&1_{2\times1}\\ 1_{3\times2} & 0_{3\times3} & 1_{3\times2}&1_{3\times1} \\ 1_{2\times2} & 1_{2\times3} & 0_{2\times2}&1_{2\times1} \\ 1_{1\times2} & 1_{1\times3} & 1_{1\times2}&0_{1\times1} \\ \end{array} \end{pmatrix}$
be a symmetric matrix whose entries are blocks(diagonal blocks entry must be zero and non-diagonal blocks entry must be one).
How to prove/disprove that all the eigenvalues of $M$ are not integer if and only if the diagonal blocks are not of the same order.
Note that the number of diagonal blocks $\geq 3$ and order of diagonal blocks$\geq 1$ .