Does it stop eventually? Sounds like you’ve ruled out the obvious, the flapper, and the bypass tube. I’ve seen some buildup in the void above the jets before and wonder if they’re almost clogged but allowing a dribble through. This may give you the impression that there’s a leak, but it is actually a blockage.

If it does stop eventually, I’d suspect it’s a partial blockage. Aside from taking the toilet outside and going pressure washer crazy, or adding draino while the bowl is upside down to soak, I’m not sure what the best way to clear that would be. Hope that helps!