Unbounded representation counts as sums of $k$ prime $k$-th powers: is $\limsup f_k(n)=\infty$? (Erdős #979)

Fix $k\ge 2$ and let $f_k(n)$ be the number of representations $$n=p_1^k+\cdots+p_k^k$$ as a sum of $k$ $k$-th powers of primes $p_1,\ldots,p_k$ (the $p_i$ need not be distinct; count ordered solutions). Is it true that $\limsup_{n\to\infty} f_k(n)=\infty$? Equivalently, for every $M$ is there some $n$ representable in at least $M$ ways as a sum of $k$ prime $k$-th powers?

Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) that $\limsup_n f_k(n)=\infty$ for all $k\ge 2$ (or for a specified open $k\ge 4$), or a proof that $\limsup_n f_k(n)<\infty$ for some $k$. ADVANCES (proof or reproducible certificate required): (a) prove $\limsup_n f_k(n)=\infty$ for the first open case $k=4$ (or any single $k\ge 4$); (b) publish/formalise a complete proof for $k=3$ (currently only claimed and unpublished); (c) a rigorous construction showing $f_k(n)\ge M$ is attained for arbitrarily large $M$; or (d) a reproducible enumeration exhibiting integers $n$ with a new record value of $f_k(n)$ for a fixed $k\ge 4$, with the explicit representations as certificate. Deliver the proof/formalization, or the enumeration code plus the record witnesses.