Maximal subgroup of join of two elementary abelian normal subgroups of prime-square order contains elementary abelian subgroup of prime-square order for odd prime

Statement
Suppose $$p$$ is an odd prime and $$P$$ is a finite $$p$$-group. Suppose $$M,N$$ are elementary abelian normal subgroups of $$P$$ of order $$p^2$$ and $$K = MN$$. Then, every maximal subgroup of $$K$$ contains an elementary abelian subgroup of order $$p^2$$.