Conjectured Laplacian spectrum of Sierpiński graphs

Let $S(n,k)$ be the Sierpiński graph, with $n,k\geq 2$, and let $f$ be the polynomial defined in the source's equation for $f$. For a polynomial $g$, write $g^j$ for its $j$-fold iterate.

Laplacian spectrum conjecture. The Laplacian spectrum of $S(n,k)$ consists of the following eigenvalues:

1. $0$ with multiplicity $1$.
2. The zeros of $f^j(k-x)$, each with multiplicity

$$\frac12\left(k^{n-j}-2k^{n-j-1}+k\right),$$

for $j=0,1,\ldots,n-1$. 3. The zeros of $f^j(k-x)+2$, each with multiplicity

$$\frac12\left(k^{n-j-1}-1\right)(k-2),$$

for $j=0,1,\ldots,n-2$.

This conjecture extrapolates the explicitly computed Laplacian spectrum for the case $n=2$ from empirical evidence; the polynomial $f$ and its iterates encode the spectral values at higher levels of the Sierpiński-type graph construction.