Jonah-Konvisser local origin for a collection of proper subgroups

Statement
Suppose $$p$$ is a prime number, and $$G$$ is a nontrivial finite $$p$$-group of order at least $$p^2$$. Suppose $$\mathcal{S}$$ is a collection of proper subgroups of $$G$$ (For our purposes, if $$\mathcal{S}$$ originally contained $$G$$ itself, we could throw it out).

A local origin for $$\mathcal{S}$$ is a proper subgroup $$A$$ of $$G$$ such that if $$N$$ is a maximal subgroup of $$G$$ containing an element of $$\mathcal{S}$$, then $$A \cap N$$ also contains an element of $$\mathcal{S}$$.

Related notions

 * Jonah-Konvisser origin for a collection of proper subgroups: It is clear from the definitions of both terms that any maximal subgroup containing a local origin is an origin.

Facts for which this is used

 * Jonah-Konvisser line lemma
 * Local origin corollary to line lemma