Follow up email after a PhD interview

Hi,

I had a PhD interview for a position in an EU university that I really liked, and envision myself to do that kind of research for my PhD (STEM). We had (or at least I thought) a good discussion during the interview that lasted almost 2hrs. And on the same week, the prospective PI also contacted my references to have a zoom call with them to discuss me, and I thought this was more serious than merely just asking reference letters.

Now, its been 4 weeks after the interview and I haven't heard anything from them. The PI said to me during the interview and in his reply for my thank-you email that they would give me more information in the "following weeks". So, I wonder if it is advisable for me to shoot up a follow up email about the decision. If yes, how should I phrase it so that I don't sound too pushy?

I've been applying to other positions too, but I really want to get this one. Thanks!