The rainbow antichain-union conjecture

Let $A_k$ be an antichain of size $k$, and let $\bigvee_s$ and $\bigwedge_s$ denote the posets consisting respectively of a common minimal or maximal element together with $s$ incomparable elements. Let $F(n,l,P)$ be the maximum size of a color class in an $l$-coloring of $B_n=2^{[n]}$ with no rainbow copy of $P$. Rainbow antichain-union conjecture. For any $k,s$ and $l\ge k+s+1$,

$$F(n,l,\vee_s+A_k)=F(n,l,\wedge_s+A_k)=o(2^n).$$

The conjecture extends the preceding bounds for disjoint unions of antichains with fork-shaped posets; no resolution is supplied in the source.