Bounding matrix-vector product norm

Question

Let $A,B : \mathbb{R}^{n} \to \mathbb{R}^{n}$ be diagonalizable matrices with $\lambda$ being the eigenvalue of maximal absolute value between them. Let $v \in \mathbb{R}^{n}$. Running a few computations on matrices I randomly picked seems to suggest that $$ \left\| \left( \prod_{i = 1}^{n} A^{a_{i}}B^{b_{i}} \right)v \right\|_{2} \leq C\lambda^{n} $$ for all $n$, where $(a_{i},b_{i}) \in \lbrace (0,1),(1,0) \rbrace$ and $C$ is a fixed constant. Could this be true? Expanding $v$ as a linear combination of eigenvectors for $A,B$ and getting a nasty nested sum got me nowhere.