Jonah-Konvisser lemma for elementary abelian-to-normal replacement for prime-cube and prime-fourth order

Statement
Suppose $$p$$ is an odd prime number, $$P$$ is a finite $$p$$-group, and $$E_1,E_2$$ are distinct elementary abelian normal subgroups, both of order $$p^k$$, with $$N = E_1E_2$$ their join. Further, suppose that either $$[N,N]$$ has order at most $$p^2$$ or $$N/Z(N)$$ has order at most $$p^2$$.

Then, each maximal subgroup of $$N$$ containing $$E_1 \cap E_2$$ contains an elementary abelian subgroup of order $$p^k$$.