Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$?

We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$.

As an example, if the abc-conjecture is true in the form $max(|a|,|b|,|c|) \leq rad(abc) ^2 $ then $$rad(2^n \pm 1) \geq 2^{n/2 - 1}.$$ I wonder if this estimate is proven (or perhaps conjectured) by anyone? Are there any nontrivial results here?