On integers with many representations as the sum of $k$th powers of primes

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{f}_k(n)=\mathrm{\infty }\mathrm{.}$$ This positively answers a conjecture of Erdos, which asks if there are natural numbers $n$ with arbitrarily many distinct representations of the form $p_1^k+p_2^k+\cdots +p_k^k$ for primes $p_1,p_2,\dots ,p_k$.