I'm with the majority - no, not worth an update if there's nothing they can look at yet. Now if you think you can collect enough data/ get your PI to sign off on a preprint being posted to BioarXiv, then yes you could send an update and include the link, but that also may not be feasible in terms of timing.
When I was applying to med school I had 2 papers I was writing that I was really excited about, but one kept being rejected (I know that doesn't sound good but I swear it's still probably the paper I'm proudest of) and the other wasn't ready to be submitted yet. I printed out the figures for both papers for my second interview, so that I could talk people through them and I really do think it helped them understand our findings and why the research was interesting/ important. If you have some tables/figures that it would be OK to screenshare if you're asked about your research during an interview, that might be one way to use the new project to your advantage even if it's not quite ready to be published.