Dimension-boundedness conjecture for cover-planar posets

A poset is cover-planar if its cover graph is planar. For a poset $P$, let $\operatorname{se}(P)$ be its standard example number, and let $\dim(P)$ denote its dimension. A class of posets is $\dim$-bounded if there is a function $f$ such that

$$\dim(P)\leq f(\operatorname{se}(P))$$

for every poset $P$ in the class.

Dimension-boundedness conjecture. The class of cover-planar posets is $\dim$-bounded.

Cover-planar posets can have arbitrarily large dimension, so the conjecture asks whether their dimension is nevertheless controlled by their standard example number. The source traces the conjecture’s first printed reference to an informal comment published in 1992 and does not state that it has been resolved.