You are right, this study was technically conducted by The Stanford Law AI Initiative which is co-chaired by Julian Nyarko who is also a senior fellow at HAI, and is also the lead author of this study.
This is enough of an association to claim a conflict of interest between the study authors and Google. But I wanted to go further and see if The Stanford Law AI Initiative had been given a research grant from HAI. So I spent way to long on both of their websites to find a list of research grants either awarded by HAI or received by Stanford Law AI Initiative. But no such luck. Despite HAI having a page dedicated to Centers and Labs, and to Research partners, and despite claiming 500+ research funded, they only list like 6 organizations each, and then link to each other in their “See More” button below.
I have a feeling I will have to browse through some tax filing papers to find the truth here. But I am not a journalist, so I am not gonna. I am simply gonna leave it at the obvious associations involved here. And maybe issue a correction: “conducted by a senior fellow at HAI”