Weak–strong consistency conjecture for stationary ergodic distributions

Let $H_0$ and $H_1$ be hypotheses consisting of stationary ergodic distributions. A test is weakly asymmetrically consistent (respectively, weakly asymptotically consistent) when the relevant convergence holds in probability, while strong consistency requires almost-sure convergence. Weak–strong consistency conjecture. If there exists a weakly asymmetrically consistent (respectively, weakly asymptotically consistent) test, then there exists a strongly asymmetrically consistent (respectively, strongly asymptotically consistent) test. The paper notes that the corresponding implication is known for independent identically distributed or strongly mixing processes, and that the notions are equivalent in the asymmetric case when $H_0=\mathcal E\backslash H_1$; the general stationary ergodic case remains open.