Factorization conjecture for the characteristic polynomial of the asymmetric annihilation Markov matrix

Let $M_L$ be the Markov matrix for a system of size $L$, with characteristic polynomial $P_L(x)$. Define

$$A_L(x)=\prod_{k=0}^{\lceil L/2\rceil}(x+2k)^{\binom{L-1}{2k}},\qquad B_L(x)=\prod_{k=0}^{\lfloor L/2\rfloor}(x+2k+1)^{\binom{L-1}{2k+1}}.$$

Characteristic-polynomial factorization conjecture. The characteristic polynomial and its successive ratios are

$$P_L(x)=A_L(x)A_L(x+2\alpha+\beta)B_L(x+\beta)B_L(x+2\alpha),$$ $$\frac{P_{L+1}(x)}{P_L(x)}=B_L(x+1)B_L(x+2\alpha+\beta+1)A_L(x+\beta+1)A_L(x+2\alpha+1).$$

If true, this would imply that the Markov matrix has only $2L$ distinct eigenvalues; the paper supports the formula by explicit calculations, but does not establish it in general.