Second-author conjecture on weak versus induced poset Ramsey numbers

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, let $R(Q_n,Q_n)$ be the induced poset Ramsey number, and let $R^{\mathrm{w}}(Q_n,Q_n)$ be the weak poset Ramsey number. Weak-versus-induced Ramsey conjecture. For any $n\in\mathbb{N}$,

$$R(Q_n,Q_n)-\Theta(n)\le R^{\mathrm{w}}(Q_n,Q_n)\le R(Q_n,Q_n).$$

The upper bound is immediate from the definitions, while the paper reports an upper bound of $0.96n^2$ for the weak number and says it remains open whether weak poset Ramsey numbers are significantly smaller than induced ones. The conjecture is therefore presented as an open suggestion by the second author.