The standard-example conjecture for high-dimensional posets

A standard example of dimension $m$ is the poset formed by the one-element subsets and the $(m-1)$-element subsets of an $m$-element set, ordered by inclusion. Standard-example conjecture. For every $t<1$ sufficiently close to $1$, there exists a constant $c>0$ such that every poset with $2n$ points and dimension at least $tn$ contains a standard example of dimension $cn$. The conjecture asks whether posets whose dimension is close to the maximum possible relative to their size must contain a linearly large standard example. The source gives no resolution.