For a positive real number $x$, denote the fractional part $x-[x]$ of $x$ by $\langle x \rangle$.

Fix a prime $p$. Is it known whether or not

$$\Theta_p := \liminf_{n>0 \text{ not a } p^{\text{th}} \text{ power}} \ \sum_{k = 1}^{p-1} \left\langle n^{\frac{k}{p}}\right\rangle$$

is zero?

EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line.

For a positive real number $x$, denote the fractional part $x-[x]$ of $x$ by $\langle x \rangle$.

Let $\ell>0$ be an integer. Is

$$\Phi_{\ell} := \liminf_{n>0 \text{ not a } {\ell}^{\text{th}} \text{ power}} \ \sum_{k = 1}^{\ell^2} \left\langle n^{\frac{k}{\ell}}\right\rangle$$

equal to zero?

Fix a prime $p$. Is it known whether or not

$$\Theta_p := \liminf_{n>0 \text{ not a } p^{\text{th}} \text{ power}} \ \sum_{k = 1}^{p-1} \left\langle n^{\frac{k}{p}}\right\rangle$$

is zero?