On boundedness of infinite matrix products with alternating factors from two sets of matrices

We consider the question of the boundedness of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to an appropriate choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ for which the sequence of matrix products $\{A_{n}B_{n}\cdots A_{1}B_{1}\}_{n=1}^{\infty}$ is norm bounded. Some situations are described in which in this case the norms of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ are uniformly bounded, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C$ for all natural numbers $n$, where $C>0$ is some constant independent of the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$. In the general case, the question of the validity of the corresponding statement remains open.