They explicitly say they do not want pre-interview updates. I don't know about letters of interest, but I guess it would follow the same rule.
Now, instructions on letters of interest/updates on interview day are in some conflict with the website and my communication with the admissions office. On interview day, I wrote down that active communication with the admissions office is encouraged and "helps your chances." I don't know if it was in specific reference to waitlist movement, but it sounded like general advice for the remainder of the cycle. Moreover, the staff mentioned updating them as a means of keeping in touch, rather than only providing "significant experiential updates" per their website.
A lot has changed since then, so I don't really know what to think anymore.