Wait, where are you seeing the link to HAI? TFA mentions something called "liftlab" which seems to be something under Stanford Law School and separate from HAI. The study has more than a dozen authors from as many different universities but HAI is not mentioned.
The leader of the study, Julian Nyarko, is Associate Director and Senior Fellow at HAI. I can't say whether that means the study was conducted by HAI, but there is at least a connection to it. https://hai.stanford.edu/people/julian-nyarko
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”