Kiselev–Kozik conjecture on satisfying sequences for rainbow partite matchings

Let $s,n,k$ be positive integers, and let a sequence $(f_i)_{i=1}^s$ be called satisfying when it has the property that any $s$ families in $[n]^k$ whose respective sizes exceed $f_i$ contain a cross-matching. The sequences

$$f_i=i n^{k-1}$$

and

$$f_i=\min\{i+C\sqrt{s\log s},s-1\}n^{k-1},\qquad i=1,\ldots,s,$$

are satisfying, where $C$ is an absolute constant. Kiselev–Kozik's conjecture. The two sequences above are satisfying. The first sequence asserts the sharp asymmetric multipartite analogue of the Aharoni–Howard conjecture, while the second captures the previously observed square-root error term. The Aharoni–Howard conjecture itself was resolved for all $s>470$, but these stronger assertions about satisfying sequences remain open in the supplied source.