We consider the behaviour of the Fisher information of scaled sums of independent and identically distributed random variables in the Central Limit Theorem regime. We show how this behaviour can be related to the second-largest non-trivial eigenvalue associated with the Hirschfeld--Gebelein--Rényi maximal correlation. We prove that assuming this eigenvalue satisfies a strict inequality, an $O(1/n)$ rate of convergence and a strengthened form of monotonicity hold.