Near-trivial additive term conjecture for fixed-poset Ramsey numbers

Let $P$ be a fixed finite poset independent of $n$, and let $Q_n$ be the $n$-dimensional Boolean lattice. The poset Ramsey number $R(P,Q_n)$ is the least dimension of a Boolean lattice whose every blue/red coloring contains a blue copy of $P$ or a red copy of $Q_n$.

Fixed-poset asymptotic conjecture. For $n\in\mathbb{N}$ and fixed $P$,

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

The source motivates this by a known upper bound of the form $C_Pn$ and a lower bound exceeding $n$ by order $n/\log n$. It also states that this conjecture is equivalent to the fixed-$m$ Boolean-lattice formulation in Candidate 10, and remains open.