Supermodular ordering of speeded-up Markov chains without stochastic monotonicity

For a stationary continuous-time Markov chain $\lambda$ with real values, let $\lambda_c(t)=\lambda(ct)$. Leskelä asked whether there is a stationary ergodic chain that is not stochastically monotone but satisfies $\lambda_{c_1}\ge_\mathrm{sm}\lambda_{c_2}$ whenever $c_1\le c_2$, where $\ge_\mathrm{sm}$ is the supermodular order of processes. We show that such chains exist on three states. If $\lambda$ jumps from its smallest value to each of the other two values at arbitrary positive rates and returns from both of them at a common rate, then $\lambda_{c_1}\ge_\mathrm{sm}\lambda_{c_2}$ for all $c_1\le c_2$, while $\lambda$ is not stochastically monotone. The proof reduces monotonicity in each sampling gap to two finite inequalities for the conditional expectations of the test function, and proves them with a coupling and an explicit decomposition into elementary increasing differences. If the largest value returns faster than the middle one, the ordering fails already for the pairs $(\lambda(0),\lambda(t))$. For reversible three-state chains with two distinct nonzero eigenvalues, these pairs decrease in the supermodular order if and only if the slow left eigenvector does not have the same strict sign at the two extreme states, and computations suggest that this condition characterizes the ordering of the whole process.