This is a follow-up question to Limit of sequence of growing matrices.

There I was considering a sequence of matrices defined by $$ K_L = \left[H\otimes I_{2^{L-2}}\right]\left[I_2 \otimes K_{L-1}\right], $$ where $\otimes$ denotes the Kronecker product, $I_n$ is the $n\times n$ identity matrix, and $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),\quad K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right). $$

I was interested in understanding the singular values of the matrices $K_L$. Experimentally I observed that the characteristic polynomial $\chi_L\in\mathbb{Q}[\lambda]$ of $K_LK_L^T$ scales with $\lambda^{2^L}$, in the sense that $\lambda^{2^L}\chi_L(1/\lambda)$ converges pointwise to a limiting function as $L$ tends to infinity.

I am now trying to generalise this observation to other matrices $H$; the easiest example I could come up with is $$ H'=\frac{1}{2}\left(\begin{array}{cccccc} 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 1 & 0 \end{array}\right),\quad K_1=\left(\begin{array}{c}1 \\ 0 \\ 0\end{array}\right). $$

Like for $H$ the rows and columns of $H'$ sum to one and each row also occurs as a column. Nonetheless I seem to be unable to find a scaling such that the characteristic polynomial of $K_LK_L^T$ based on this larger matrix $H'$ converges to a limit.

My question is therefore:

Does such a scaling exist? If so, how can I find it for more general matrices $H$?

**Edit:**

Even for the (apparently simpler) $4\times4$ initial matrix $$ H=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1/2 & 1/2\\ 0 & 0 & 1/2 & 1/2 \end{array}\right),\quad K_1=\left(\begin{array}{c}1 \\ 0\end{array}\right) $$ I can't find the correct scaling for the characteristic function. Any advice on this problem?