Axenovich–author near-linear conjecture for off-diagonal Boolean lattices

Let $Q_m$ and $Q_n$ be Boolean lattices of dimensions $m$ and $n$, with $n\ge m$. For every $\varepsilon>0$, consider a threshold $m_0$ beyond which the dimensions are sufficiently large.

Axenovich–author conjecture. For any $\varepsilon>0$, there is a large enough $m_0$ such that for any $m,n\in\mathbb{N}$ with $n\ge m\ge m_0$,

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

The source describes this as stronger than the Lu–Thompson conjecture. It is intended to give a near-linear bound when both Boolean-lattice dimensions grow, and remains open.