Is there any progress on a “baby abc conjecture” where you restrict attention to rational approximations of Nth roots?

Let (r/s) be a very close approximation to (t/u)^(1/n), so that |ur^n - ts^n| = a is small. we have a pseudo-abc-triple with c being the larger of ur^n and ts^n, and q=log(c)/log(arstu).

Example: the fifth root of 109 is very close to 23/9, so we get 2+109(9^5)=23^5, 5log(23)/log(45126) = 1.4628, not quite as good as the 1.6299 it gets as an abc triple where you use that 9 = 3^2 but still good enough to make it the “best” known approximation of any root.

Is anything known about q in terms of n?