For the following class of matrices, are the determinants invariant under permutations?

Question

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-zero elements at corners.) $$ \begin{bmatrix} \Gamma_{1,1} \cdot \mathcal{I} & \Gamma_{1,2} \cdot P_{1} & 0 & \underline{0} & \Gamma_{1,m} \cdot P_{m} \\ \hline \Gamma_{1,2} \cdot P_{1}^{T} & \Gamma_{2,2} \cdot \mathcal{I} & \Gamma_{2,3} \cdot P_{2} & \underline{0} & 0 \\ \hline 0 & \Gamma_{2,3} \cdot P_{2}^{T} & \Gamma_{3,3}\cdot \mathcal{I} & \Gamma_{3,4} \cdot P_{3} & \underline{0} \\ \hline \underline{0}^{T} & \ddots & \ddots & \ddots & \ddots \\ \hline \Gamma_{1,m} \cdot P_{m}^{T} & 0 & 0 & \cdots & \Gamma_{m,m}\cdot \mathcal{I} \end{bmatrix} $$ where $\{P_{i}\}_{i=1}^{m}$ are the permutation matrices with size $M \times M$.

I was trying to prove it by using "Newton's identities" that can connect the determinant with the traces of the n-th power of the matrix. Showing that the diagonal blocks of the $n$th power of this matrix are always identity matrices up to some factors gives us the result.

I didn't find a systematic way to show that the $n$th power of the above matrix has the same diagonal blocks. Or was my intuition wrong?

In this problem, we have two parameters: the size of the base matrix $m$ and the size of permutation matrices $M$. (Perhaps the invariance can be established in some region $(m, M)$.）

(Sorry for the repeated question since I have not enough reputation to give any comments on relevant questions. see also For the following class of matrices, are the determinants invariant under permutations? on MSE.)

Based on Lemma 1 in Molinari - Determinants of block tridiagonal matrices, I can easily come up with this result by forcing $\bigl(\prod\limits_{i=1}^{m-1} P_{i}\bigr) \cdot P_{m}^{T} = \mathcal{I}$ (i.e., $P_{m} = \prod\limits_{i=1}^{m-1} P_{i}$). (Maybe it can also come up with the result without such a constraint.)

The determinant will be equivalent to the determinant of the matrix with the following form: $$ \begin{bmatrix} t_1 \cdot \mathcal{I}- h \cdot \bigl(\prod\limits_{i=1}^{m-1}P_{i}\bigr)\cdot P_{m}^{T} & t_2 \cdot P_{m} \\ t_3 \cdot \bigl(\prod\limits_{i=1}^{m-1}P_{i}\bigr)^{\mathrm T} & t_4 \cdot \bigl(\prod\limits_{i=1}^{m-1}P_{i}\bigr)^{T}\cdot P_{m} - \mathcal{I} \end{bmatrix} $$ Trivially, we can get its invariance under permutations with the constraint that $\bigl(\prod\limits_{i=1}^{m-1} P_{i}\bigr) \cdot P_{m}^{T} = \mathcal{I}$.

Given $\{P_{i}\}_{i=1}^{m-1}$ (say, $P_{i} = \mathcal{I}$, $\forall i \in [m-1]$), if we can prove that their determinants will be unchanged if we consider different $P_{m}$, then we can conclude that determinants are invariant under permutations.

In the above situation, the determinant can be written as $\det((t_1\cdot t_4 - t_2 \cdot t_3) \cdot P_m - h\cdot P_m^{T} - (t_1+h\cdot t_4)\cdot \mathcal{I}$), which, I think, is dependent on $P_m$. Therefore, the statement is not true for all permutations (?).

It seems that the determinant can be eventually reduced to be in the following form: $\det(\ell_{1}\cdot \mathcal{I} + \ell_{2}\cdot W + \ell_{3} \cdot W^{T})$, denoted by $D(W)$, where $W$ is an arbitrary permuation matrix with size $M \times M$. When $W = \mathcal{I}$, we denote the determinant as $D_{\mathrm{id}}$. Based on this observation, we can reduce the number of possible determinants from $\left(M!\right)^{m}$ to $M!$ and we can conduct numerical simulations for $M < 10$. The simulation result shows that the average of $D(W)$ with respect to $W$, denoted by $\langle D(W)\rangle$, are equal to $D_{\mathrm{id}}$ for $M < 10$.

Equivalently, is there any nice way to discuss the following class of matrices: $$ \begin{bmatrix} \Gamma_{1,1} \cdot \mathcal{I} & \Gamma_{1,2} \cdot \mathcal{I} & 0 & \underline{0} & \Gamma_{1,m} \cdot P_{m} \\ \hline \Gamma_{1,2} \cdot \mathcal{I} & \Gamma_{2,2} \cdot \mathcal{I} & \Gamma_{2,3} \mathcal{I} & \underline{0} & 0 \\ \hline 0 & \Gamma_{2,3} \cdot \mathcal{I} & \Gamma_{3,3}\cdot \mathcal{I} & \Gamma_{3,4} \cdot \mathcal{I} & \underline{0} \\ \hline \underline{0}^{T} & \ddots & \ddots & \ddots & \ddots \\ \hline \Gamma_{1,m} \cdot P_{m}^{T} & 0 & 0 & \cdots & \Gamma_{m,m}\cdot \mathcal{I} \end{bmatrix} $$ where $P_{m}$ is an arbitrary permutation matrix with size $M \times M$.