Local origin corollary to line lemma

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 any subgroup $$H$$ of $$G$$, define $$n(H)$$ as the number of subgroups of $$H$$ in $$\mathcal{S}$$.

Consider the statement $$A(H)$$ for a subgroup $$H$$ of $$G$$:

$$A(H)$$: Either $$n(H) = 0$$ or $$n(H) \equiv 1 \pmod p$$.

Suppose there exists a local origin for $$\mathcal{S}$$ in $$G$$: a proper subgroup $$A$$ of $$G$$ such that for every maximal subgroup $$N$$ of $$G$$ that contains an element of $$\mathcal{S}$$, $$A \cap N$$ also contains an element of $$\mathcal{S}$$. Then, we have the following:

If $$A(M)$$ holds for every maximal subgroup $$M$$ of $$G$$, $$A(G)$$ holds.

Facts used

 * 1) uses::Jonah-Konvisser line lemma

Proof
Any maximal subgroup containing a local origin is an origin, so in particular, the existence of a local origin implies the existence of an origin. Then, fact (1) yields the desired result.