Non-commuting transformations with non-converging 2-fold ergodic averages

In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with $\mathrm{\mid }{N}_{m+1}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{N}_m\mathrm{\mid }\to \mathrm{\infty }$ such that $T^{i}A\cap S^{i}A=\phi$ for $i\in [N_{4k},N_{4k+1}]$ and $T^{i}A=S^{i}A$ for $i\in [N_{4k+2},N_{2k+3}]$. For the corresponding deterministic Gaussian and Poisson suspensions $S_{\circ },T_{\circ }$ over $S,T$ for some $f\in L_{\mathrm{\infty }}$ there is no limit of $\frac{1}{N}{\sum }_{n=1}^N{T}_{\circ }^{n}f{S}_{\circ }^{n}f$ in $L_2$.