Doing part-time internship before starting full-time at another firm?

I accepted a FT offer at a HF that I will start next summer. I'm currently still in school and considering taking a part-time semester internship at a startup. Rationale behind doing this- I'm interested in learning more about startups/tech, and it's a way to earn some money while doing pretty interesting work. The startup is small (3 people), not well known, in an industry different from the one I'll be covering at the HF. The startup knows that I have a FT job lined up, and neither of us have the intention to convert this to a FT job.

Are there any potential conflicts here, and is this something I need to ask/notify the HF about? I didn't think so at first, but now I'm not so sure.

This HF job is my "dream job," so I would definitely let the internship go if there was a conflict, but if there's no issue, I'd rather not miss out on what seems like an interesting opportunity. And I'd rather not contact the HF about this unless necessary because they're a conservative firm and I don't want them to think I've become interested in something completely different career-wise.

For what it's worth, I did a part-time internship all year last year, months after having accepted my summer offer at another firm, and the issue never occurred to me then. And is the situation really different from taking a semester work-study or service job?

Any guidance would be appreciated!