Diagonal-entry formula conjecture for the characteristic polynomial of the asymmetric annihilation Markov matrix

Let $M_L$ be the Markov matrix of the asymmetric annihilation model, with rows and columns indexed by $i=0,\ldots,2^L-1$. For an integer $m$, define $n_1(m)$ to be $(-1)$ raised to the number of ones in the binary expansion of $m$. Diagonal-entry characteristic-polynomial conjecture. The characteristic polynomial of $M_L$ is

$$P_L(x)=\prod_{i=0}^{2^L-1}\left(x-(M_L)_{(i,i)}-\alpha n_1(i)\right).$$

This is presented as an alternative form of the conjectured characteristic polynomial and is not proved in the supplied text; its validity would give the spectrum directly from the diagonal entries of $M_L$ and the binary-parity function $n_1$.