 Solution

Let $L$ be the number of index tables, including the zero table. If distinct feasible states $A,B$ differ by an index table, then $P(A,B)=1/L=P(B,A)$, using the sign-reversal bijection for the reverse move. If they do not differ by an index table, both probabilities are zero. Since $\pi$ is uniform,For equal states detailed balance is tautological. Thus the chain is reversible and its uniform distribution is stationary.