And here I was getting ready to say something nice to say about it.
After reading the supplement I don’t think it is BS, but I’m not convinced it is useful either, at least not yet. Then my weekend got busy, so I haven’t followed up the references.
The basic framework for the Assembly Index (AI) looks OK to me. The title seems like a grand claim, but if we set biology aside, I think the “explains and quantifies selection” part is OK. If we understand “and evolution” to mean adding more parts as the paper describes, again without biology, that might be OK too.
Can this be usefully applied to biology? I still don’t know.
There is a citation to a paper an algorithmic information theory which I am familiar with (Wallace 1999). I think AI and Minimum Message Length (MML*) are related in how they reduce the data to essential parts.
*MML is an important statistic if you want to make statistical inferences using Algorithmic Information. Nobody does that because it doesn’t work very well, or so I’ve read. I don’t think this bodes well for applications of AI either, but that remains to be seen.
Note: I’m pretty sure Dembski (2005) cited this same paper by Wallace. Dembski didn’t actually use MML, but it would be a wonderful irony if MML or something like it went on to be a key to explaining biological evolution. 
Another citation is to is an article on Constructor theory (Deutsch 2015), which is a topic I’ve been trying to follow. Constructor theory requires enumerating all the states a system can change (or evolve) into, and the AI seems like a step toward doing that.
I will need to look closer at the connection between MML and AI, and the other references. Stay tuned.