Biró–Füredi–Jahanbekam stability conjecture for standard examples in posets

Let $P$ be a finite poset, let $|P|$ denote its number of elements, and let $\dim(P)$ denote its order dimension. For a positive integer $m$, let $S_m$ be the standard example on $2m$ elements, consisting of the $1$-element and $(m-1)$-element subsets of an $m$-element set ordered by inclusion. Biró–Füredi–Jahanbekam conjecture. For every $t<1$, but sufficiently close to $1$, there are a real number $c>0$ and a positive integer $N$ such that if

$$|P|\geq 2n\geq N \quad\text{and}\quad \dim(P)\geq tn,$$

then $P$ contains $S_{\lfloor cn\rfloor}$ as a subposet. This is a stability conjecture for Hiraguchi's bound: posets whose dimension is close to the maximum possible should contain a standard example whose size is linear in the number of elements. The source attributes the conjecture to Biró, Füredi, and Jahanbekam; no resolution is given here.