For a positive integer $k$ let $\gamma_k(n)$ be the number of representations of $n$ as a sum of strictly increasing perfect $k^{\text{th}}$ powers. For example $\gamma_k(2)=0$ for any $k$. Now is the following true?

For any $x\in \mathbb{R}^\ast_+$ there's an integer $N$ such that for all $n\ge N$ $$\dfrac{\max_{1\le i\le n}\;\gamma_k(i)}{n}>x$$

Motivation : This is motivated from an easier problem : Let $\gamma^{\ast}_k(n)$ be number of representations of $n$ as sum of distinct $k^{\text{th}}$ powers, where order doesn't matter. Then $\exists \; n$ such that $$\gamma_k^{\ast}(n)>nx$$ for any positive real $x$. Can someone shed some light on this conjecture of mine? I have tried to check as much as possible,and I have found no way out of it. Thanks for all help.