Don't Think I'm On track for a return offer

Hi all,

Hope everyone's having a good day so far! Interning FO at a BB, and over half of the internship has gone by... there are some other interns in my pod who I think have been doing a "better" job than me so far, and I don't think I'm on track to receive a return offer.

I put better in quotes b/c they're more visible; they do more coffee chats and are better at socializing/ingratiating themselves with people, and because of this, get the "sexier" projects. Ofc, because they do so many coffee chats, I'm left with the brunt of the group projects (which are more along the lines of tedious work). I don't want to confront them about it or the manager because I think it's pretty obvious whose side they'll take, and I don't want to cause conflict when there's only a few weeks left of the internship.

Now me, personally, I've been doing all the work I've been assigned within the deadlines, and trying to ask for the more interesting projects but have been blown off most of the time even though they tell me my work quality is sound. Headcount at this office is kinda (?) limited so yeah I can definitely see the writing on the wall - would be blind to not notice it. My question is, would they be receptive of writing me a rec letter to other full time jobs once this internship is over? I haven't shit the bed, I just haven't been as "visible" as the other interns. And do I start reaching out on LinkedIn now in advance, while I'm still in the middle of this internship? Would that look too self-serving/ be in bad taste?

What would your advice be for me, if any of you all have ever been in the same position? Do I just focus on networking now with people from the firm who have worked at other banks, and try to go from there? While maintaining the status quo and praying that they'll write me a rec letter to other places? Not trying to burn bridges, but I think my facial expression might've shown that I was annoyed at things like just once or twice because its just... too much sometimes haha

Looking for advice, and throwaway acc for obvious reasons. Thank you guys in advance!