Can a list with $\mathcal{O}(1)$ access have an insertion complexity better than $\mathcal{O}(n)$?

Question

It seems intuitive that there's no list data structure which has $\mathcal{O}(1)$ worst case time complexity for random access and a worst case complexity better than $\mathcal{O}(n)$ for insertion: if insertion is allowed to affect only a small part of the list, then there's no way for it to always keep the entire list in a structure that allows for constant time element access.

Is there any proof (or counterexample) of this?