Fixed-dimensional diagonal poset Ramsey asymptotic conjecture

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, and let $R(Q_m,Q_n)$ be the induced poset Ramsey number. Fixed-$m$ asymptotic conjecture. For fixed $m\in\mathbb{N}$ and large $n\in\mathbb{N}$,

$$R(Q_m,Q_n)=n+o(n).$$

This conjecture is attributed in the source to the authors of the cited reference and concerns the asymptotic behavior when the smaller Boolean lattice is fixed. The source gives no resolution, so it remains open.