Jonah-Konvisser 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).

An origin for $$S$$ is a maximal subgroup $$M$$ of $$G$$ such that if $$N$$ is another maximal subgroup of $$G$$ containing an element of $$\mathcal{S}$$, then every maximal subgroup of $$G$$ containing $$M \cap N$$ also contains an element of $$\mathcal{S}$$.

Related notions

 * Jonah-Konvisser local origin for a collection of proper subgroups: Being a local origin is a fairly strong condition that implies that every maximal subgroup containing it is an origin.

Facts for which this is used

 * Jonah-Konvisser line lemma