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?

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

Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that $$ |u\cdot r^n - t\cdot s^n| = a $$ is small. We thus have a pseudo-$abc$-triple with $c$ being the larger of $u\cdot r^n$ and $t\cdot s^n$, and $q=\frac{\log c}{\log(arstu)}$.

Example: the fifth root of $109$ is very close to $23/9$, so we get $$ 2+109\cdot 9^5=23^5,\quad 5\log(23)/\log(45126) = 1.4628, $$ 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 $n$?