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It is shown that for polynomials $p_1, p_2 \in {\mathbb Z}[t]$ with ${\rmdeg}\ p_1, {\rm deg}\ p_2\ge 5$ there exist a probability space $(X,{\mathcalX},\mu)$, two ergodic measure preserving transformations $T,S$ acting on$(X,{\mathcal X},\mu)$ with $h_\mu(X,T)=h_\mu(X,S)=0$, and $f, g \inL^\infty(X,\mu)$ such that the limit \begin{equation*}  \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{p_1(n)}x)g(S^{p_2(n)}x)\end{equation*} does not exist in $L^2(X,\mu)$, which in some sense answers aquestion by Frantzikinakis and Host.

A counterexample on polynomial multiple convergence without commutativity