Subquadratic Boolean lattice Ramsey number conjecture

Let $Q_m$ and $Q_n$ denote Boolean lattices of dimensions $m$ and $n$, respectively, and let $R(Q_m,Q_n)$ be the least integer $N$ such that every red-blue coloring of the Boolean lattice $Q_N$ contains either a red copy of $Q_m$ or a blue copy of $Q_n$. Assuming, without loss of generality, that $n\geq m$, Subquadratic Ramsey number conjecture. For sufficiently large $m$ and $n$,

$$R(Q_m,Q_n)=o(n^2).$$

There remains a significant gap between the paper's upper bounds and the best known lower bounds, so the authors expect the true values to be significantly smaller than their upper bounds.