The bounded-size optimal local realizer conjecture

Let $P$ be a finite poset. A local realizer of $P$ is a family of partial linear extensions whose frequency is the maximum number of partial linear extensions in which any element of $P$ appears, and whose size is the number of partial linear extensions in the family. Write $\operatorname{ldim}(P)$ for the minimum possible frequency of a local realizer of $P$, and $\dim(P)$ for the Dushnik–Miller dimension of $P$. The conjecture asserts the following. Bounded-size optimal local realizer conjecture. There exists a function $f$ such that for every poset $P$, there is a local realizer $\mathcal{L}$ of $P$ with frequency exactly $\operatorname{ldim}(P)$ and size at most $f(\dim(P))$. This asks whether an optimal local realizer can always be chosen with size controlled solely by the ordinary dimension of the poset, despite the fact that local realizers can generally be much larger than their frequency. The source describes this as a very bold conjecture and gives no resolution.