Weak versus induced poset Ramsey numbers conjecture

Let $Q_n$ be the $n$-dimensional Boolean lattice. Write $R(Q_n,Q_n)$ for the induced poset Ramsey number and $R^{\mathrm{w}}(Q_n,Q_n)$ for the weak poset Ramsey number, in which the monochromatic copy need not be induced.

Weak-versus-induced conjecture. For every $n\in\mathbb{N}$,

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

The upper inequality is immediate from the definitions, while the conjecture asserts that the weak and induced diagonal Ramsey numbers differ by at most a linear additive term. The source states that whether the weak number is significantly smaller remains open.