  In this note we introduce a sequence of bilinear operators that unify ergodic
averages and backward martingales in a nontrivial way. We establish its
convergence in a range of $L^p$-norms and leave its a.s. convergence as an open
problem. This problem shares some similarities with a well-known unresolved
conjecture on a.s. convergence of double ergodic averages with respect to two
commuting transformations.
