Lower Estimate on Square Function of an Indicator Set

Let $Sf$ be a discrete martingale square function. Then, for any set $V$ of positive probability, we have $\mathbb{E}S({\mathbf{1}}_V{)}^2\ge \eta \mathbb{P}(V)$ for an absolute constant $\eta >0$. We extend this to wavelet square functions, and discuss some related open questions.