# Product of random matrices which commute almost surely

In the paper “Matrix concentration for products” it is stated, that the following is easy to show.
Let $$X_1,dots X_n$$ be independent, bounded, square matrices, which commute almost surely. Define $$Y_i=I+frac{X_i}n$$. Then
$$log mathbb{E}|Y_ndots Y_1|leq frac 1 n |sum_{i=1}^nmathbb{E}X_i| +Oleft(sqrt{frac{log d}n}right)$$
$$|cdot|$$ is the spectral norm and $$d$$ is the dimension of our matrices.
I know the weaker inequality for matrices which do not necessarily commute almost surely with $$sum_{i=1}^n|mathbb{E}X_i|$$ instead of $$|sum_{i=1}^nmathbb{E}X_i|$$.
Is this really easy to show? How do I prove it?