On the 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, $A_i\in \mathcal{A}$ and $B_i\in \mathcal{B}$, 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 $\{A_n{B}_n\cdots A_1{B}_1{\}}_{n=1}^{\mathrm{\infty }}$ 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, $\mathrm{\parallel }{A}_n{B}_n\cdots A_1{B}_1\mathrm{\parallel }\le C$ 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.