I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that

$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,$$

where $\|\cdot\|_\infty$ denotes the operator norm, and $\hat{A_i}$ denotes the omission of $A_i$.

E.g. if all products of subsets of $\{A_1,\ldots,A_k\}$ are normal, then the above inequality should follow from the submultiplicativity of the operator norm and the fact that $AB$ and $BA$ have the same singular values for normal $A$ and $B$. However, this condition seems rather artifical and I am wondering if something more powerful holds. What if all $A_i$ are normal, self-adjoint, unitary or orthogonal projectors?