  We aim to showcase the wide applicability and power of the big pieces and
suppression methods in the theory of local $Tb$ theorems. The setting is new:
we consider conical square functions with cones $\{x \in \mathbb{R}^n \setminus
E: |x-y| < 2 \operatorname{dist}(x,E)\}$, $y \in E$, defined on general closed
subsets $E \subset \mathbb{R}^n$ supporting a non-homogeneous measure $\mu$. We
obtain boundedness criteria in this generality in terms of weak type testing of
measures on regular balls $B \subset E$, which are doubling and of small
boundary. Due to the general set $E$ we use metric space methods. Therefore, we
also demonstrate the recent techniques from the metric space point of view, and
show that they yield the most general known local $Tb$ theorems even with
assumptions formulated using balls rather than the abstract dyadic metric
cubes.
