Consider the following setup: Let $\{S_n, \mathcal{F}_n: n \geq 1\}$ be a martingale on a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$, with $S_0 = 0$ and differences $X_n = S_n - S_{n-1}$ for $n = 1, 2, \ldots$
Define the following: $$ \sigma_n^2 = E(X_n^2 | \mathcal{F}_{n-1}), \quad V_n^2 = \sum_{i=1}^n \sigma_i^2, \quad s_n^2 = E V_n^2 $$
In all the martingale CLT literature I've seen (such that this one), there's usually some requirement like:
(1) Throughout, we consider martingales for which $V_n^2 s_n^{-2} \to 1$ in probability as $n \to \infty$.
And then under this LLN-like condition for the conditional variance, the final result is that the martingale $S_n$, when normalized by $s_n$, is asymptotically normal.
My question: what happens if instead of norming by $s_n$, I use $V_n$ instead? I'm looking at a particular application where I actually can compute $V_n$ but not $s_n$. Does using $V_n$ break something fundamental about the CLT and, if so, is there an easy way to see it?