Monochromatic subdiagram cover-dimension conjecture

Let $P$ be a poset, let $D$ be its order diagram, let $E$ be the edge set of $D$, and let $\phi:E\to\{1,2,\dots,r\}$ be an $r$-coloring of the edges. For a subdiagram $D'$ with associated suborder $P'$, let $\dim_{uc}(D')$ and $\dim_{lc}(D')$ denote the upper- and lower-cover dimensions used in the paper. Monochromatic subdiagram cover-dimension conjecture. For every pair $(d,r)$ of positive integers, there is a poset $P$ with $\dim(P)=d$ such that some $\alpha\in\{1,2,\dots,r\}$ satisfies, for

$$D'\text{ having edge set }\{e\in E:\phi(e)=\alpha\},$$

with $P'$ the suborder determined by $D'$,

$$\dim_{uc}(D')=\dim_{lc}(D')=\dim(P).$$

The conjecture is presented as stronger than the preceding doubling conjecture and is intended to provide a robust monochromatic-subdiagram formulation; the supplied text gives no resolution.