The integer-numerator conjecture for the stationary distribution of the finite k-chain

Let $k\geq 2$, let $\mathcal{R}_k$ be the state space of the finite $k$-chain, let $\pi$ be its stationary distribution, and set

$$M_k=\prod_{j=1}^k\binom{2j}{j}.$$

For each $\lambda\in\mathcal{R}_k$, write $A_\lambda$ for an integer. Integer-numerator conjecture. For every $\lambda\in\mathcal{R}_k$, we have

$$\pi(\lambda)=\frac{A_\lambda}{M_k}.$$

This conjectures that all stationary probabilities have the common denominator $M_k$ after clearing denominators, based on computations for $k\leq 6$.