Common divisors of a^n-1 and b^n-1 over function fields

Ailon and Rudnick have shown that if $a,b\in C[T]$ are multiplicatively independent polynomials, then $\mathrm{deg}(\gcd (a^n-1,b^n-1))$ is bounded for all $n\ge 1$. We show that if instead $a,b\in F[T]$ for a finite field $F$ of characteristic $p$, then $\mathrm{deg}(\gcd (a^n-1,b^n-1))$ is larger than $Cn$ for a constant $C=C(a,b)>0$ and for infinitely many $n$, even if $n$ is restricted in various reasonable ways (e.g., $gec(n,p)=1$).