Boundedly finite conjugacy classes of tensors

Let $n$ be a positive integer and let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. Set $T_{\otimes}(G) = \{g \otimes h \mid g,h \in G\}$. We prove that if the size of the conjugacy class $\left |x^{\nu(G)} \right| \leq n$ for every $x \in T_{\otimes}(G)$, then the second derived subgroup $\nu(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.