  It is shown that there exist a probability space $(X,{\mathcal X},\mu)$, two
ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal
X},\mu)$ with $h_\mu(X,T)=h_\mu(X,S)=0$, and $f, g \in L^\infty(X,\mu)$ such
that the limit \begin{equation*}
  \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x)
\end{equation*} does not exist in $L^2(X,\mu)$.
