bad to go swimming after sustaining a cut above the eye?

The cut doesn’t seem deep, stopped bleeding within 10 minutes. I have since applied antiseptic and a bandaid. However, I want (or need) to go swimming today, within the hour. I’ve heard chlorinated water can be good for cuts elsewhere on the body but bad on eye and mouth sores. However, I don’t have a sore near the eye, just a cut. I plan to swim for 4 hours. Can anything bad happen, like water flowing into my head from the opening?

oh and it hasn’t scabbed over yet.