An obstruction like ETH We know under $ETH$ we cannot solve $K$-SUM in $f(K)poly(nK)$ time under any function $f(K)$ (usually $2^{O(K)}$).

Is there any conjecture that prevents a $(\log n)^{O(K)}$ complexity (this is entirely consistent with possibility as $K=\Omega(n)$ we need exponential time for subset sum) or is such possibility allowed?
ETH itself precludes this possibility.

In https://people.csail.mit.edu/rrw/cnf-sat-feasible.pdf
 we show that any $n^{O(1)} n^{k/\alpha(k)}$ time algorithm for k-SUM, for any monotone nondecreasing unbounded function $\alpha$, would imply ETH is false.