If $n\geq 2$, is the function $$ f_n(a,b) := \left( \frac{\binom{n}{a}}{\binom{n}{b}}\right)^{\frac{1}{b-a}}$$ injective over the set $\{ (a,b) \;|\; 0\leq a<b \leq n, \; a+b \neq n \}$?

I heard about this problem by way of a colleague who attributed it to William Wu of Stanford. According to our mutual friend, it came up in a study of self-intersections of bezier curves. If pressed, I could perhaps produce a reference.

A few simple comments, which I'm not sure are helpful. First, it's easy to see that the functions are injective and equal to rational numbers over the subdomain where $b-a=1$, and (apparently) equal to irrational algebraic numbers over the rest of the domain. My efforts at proving this conjecture were focused in that direction. Second, I computer-verified this for $n \leq 675$.