Density of sums of $k$-th powers: is $f_{k,k}(x)\gg x^{1-\epsilon}$ and $f_{k,m}(x)\gg x^{m/k}$? (Erdős #323)

For integers $1\leq m\leq k$, let $f_{k,m}(x)$ denote the number of integers $\leq x$ that can be written as a sum of $m$ many nonnegative $k$-th powers. (i) Is it true that for every $\epsilon>0$, $$f_{k,k}(x)\gg_\epsilon x^{1-\epsilon}?$$ (ii) Is it true that for $m<k$ and all sufficiently large $x$, $$f_{k,m}(x)\gg x^{m/k}?$$

Acceptance. FULLY RESOLVES: a proof (machine-checkable Lean/Coq preferred, else a complete written proof) establishing (i) $f_{k,k}(x)\gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$ (for $k\geq 3$) and (ii) $f_{k,m}(x)\gg x^{m/k}$ for all $m<k$ and large $x$, or a disproof of either. ADVANCES: (a) prove that $f_{k,k}(x)=o(x)$ FAILS for some $k\geq 3$ (i.e. $f_{k,k}(x)\gg x$), or conversely prove $f_{k,k}(x)=o(x)$, with proof; (b) prove a power lower bound $f_{k,m}(x)\gg x^{\alpha}$ with $\alpha$ strictly larger than the best exponent stated in the background for the relevant $(k,m)$ (for $m<k$, moving toward $m/k$; the $m=3,k=3$ record is Wooley's $0.917\cdots$, cf. Erdős #325), with proof; (c) prove the exact conjectured exponent for any single new pair $(k,m)$, with proof; (d) rigorously compute $f_{k,m}(x)$ over a large range and report a certified empirical exponent with the sieving code (evidence, not a proof). Deliver the proof, the improved-exponent proof, or the sieving code plus certified counts.