This question is an attempt to find a version of mathoverflow.net/questions/190467/union-of-conjugates-in-p-groups with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite $p$-groups.

Fix a prime number $p$. Is there a sequence $\{a_k\}_{k \in \mathbb{N}}$ of real numbers with $$\lim_{k \to \infty} a_k = 0$$ 

such that for any finite $p$-group $G$, any $k \in \mathbb{N},$ any subgroup $H \leq G$ with $[G : H] \geq k$, and any subgroup $L \leq G$ such that $$L \subseteq \bigcup_{g \in G} gHg^{-1}$$ we have that $$\frac{|L|}{|G|} \leq a_k ?$$