The sharp exponent conjecture for rainbow antichain colorings

Let $A_k$ be an antichain of size $k$, and let $f(n,c,A_k)$ be the maximum size of a color class in a $c$-coloring of $B_n=2^{[n]}$ avoiding a rainbow copy of $A_k$. Sharp exponent conjecture. For any integers $(l-1)(k-1)<c\le l(k-1)$,

$$f(n,c,A_k)=2^{(1/l+o(1))n}.$$

The conjecture says that the construction described immediately beforehand has the correct exponential order for every indicated range of the number of colors; the source gives no proof of the matching upper bound.