Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:

Can we prove that $\lambda(2^n - 1) \ll 2^{o(n)}$ for infinitely many positive integers $n$?

Thanks for any advice/reference.