Basically, the argument is that horizontal gene transfer must have been such a huge part of life at its inception that the notion of a "LUCA" is confused, and we must expect to find a disgusting mess at the root of the tree of life.
Horizontal gene transfer is underestimated by biologists, but I am not sure in what sense LUCA is confused. I would not call the root of the tree of life a disgusting mess, just fuzzy.
If horizontal gene transfer was common and rampant at the outset of life, then a fundamental fact of evolutionary descent was not true at that point. This means there may NOT be a LUCA, but several progenitor organisms that each contributed to a portion of eventual life. E.g. there are four features ABCD. We may have descendants that are AB, BC, CD, AD, without having any creature that was ABCD.
Horizontal gene transfer is still rampant - actually it is a far better way to mix genes than sexual reproduction and it is probably why microbes rule the world.
LUCA is an imaginary organism that captures the features that all known living organisms share - almost certainly the history of decent of each gene in modern organisms is different.
Basically, the argument is that horizontal gene transfer must have been such a huge part of life at its inception that the notion of a "LUCA" is confused, and we must expect to find a disgusting mess at the root of the tree of life.