A note on the Poor-Verdú upper bound for the channel reliability function

In an earlier work, Poor and Verdú established an upper bound for the reliability function of arbitrary single-user discrete-time channels with memory. They also conjectured that their bound is tight for all coding rates. In this note, we demonstrate via a counterexample involving memoryless binary erasure channels (BECs) that the Poor-Verdú upper bound is not tight at low rates. We conclude by examining possible improvements to this bound.