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?