Existence of a non-modest poset

Let $P$ be a fixed non-trivial finite poset, and let $Q_n$ be the $n$-dimensional Boolean lattice. Call $P$ modest if

$$R(P,Q_n)=n+\Theta\left(\frac{n}{\log n}\right).$$

Non-modest poset conjecture. There is a fixed poset $P$ such that

$$R(P,Q_n)=n+\omega\left(\frac{n}{\log n}\right).$$

Known results establish the lower bound $R(P,Q_n)=n+\Omega(n/\log n)$ for every non-trivial $P$, and show that several classes are modest. Whether any non-modest poset exists remains open.