Subpolynomial-factor conjecture for diagonal poset Ramsey numbers

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, let $R(Q_m,Q_n)$ be the induced poset Ramsey number, and let $\varepsilon>0$. Subpolynomial-factor conjecture. There is a large enough $m_0$ such that for any $m,n\in\mathbb{N}$ with $n\ge m\ge m_0$,

$$R(Q_m,Q_n)\le n\cdot m^{\varepsilon}.$$

This is proposed as a stronger version of the Lu–Thompson subquadratic conjecture. The source gives no resolution, so the conjecture remains open.