Asymptotic Boolean Ramsey number conjecture for the two-element Boolean poset

For integers $k\geq 1$, let $R_k(B_2)$ be the least $N$ such that every coloring of the Boolean lattice $\mathcal{B}_N$ with $k$ colors contains a monochromatic copy of the Boolean poset $B_2$. Asymptotic Boolean Ramsey conjecture. For all $k\geq 1$,

$$R_k(B_2)=(2+o_k(1))k.$$

This open problem concerns the asymptotic growth of the Boolean Ramsey number for $B_2$. The paper records the lower bound $R_k(B_2)\geq 2k$ and notes that the conjectured asymptotic equality remains open.