I'm really not that well versed, but disorder and complexity seem to be used interchangeably which is incorrect:
"According to this fundamental law, entropy - and with it disorder - should always increase with time. This is manifestly not happening in the universe"
I'm completely baffled that they'd put that in their abstract or what they could possibly be thinking. In what way is entropy and disorder not increasing? And I think here is where they are confusing disorder and complexity. For a given system, as entropy initially increases, so does complexity, and then at some point, complexity peaks and falls.
A system with very low entropy is not complex. It is very easy to describe. Additionally, a system with really high entropy is also not complex, it is also very easy to describe.