Should I contact the team directly? [Interview help needed!]

Hi all,

I'm in the process of interviewing for an associate role in a sell side boutique. Currently I've passed stock pitch stage and been told the next stage will be a case study/lunch. It's been over 3 weeks and recruiter said he has not heard back re either dates or more info. I really wanted this role and am getting a bit anxious. So I was hoping that I can get your advice on:

a) whether I should contact the team directly and ask for an update (I have the contact of the analyst)?

b) what to expect in a case study? writing sample? modelling test? another pitch?

Thanks all in advance.