Multiplicative lower-bound conjecture for dimension of cover-planar posets

Let $P$ be a cover-planar poset, let $m(P)$ be the number of minimal elements of $P$, and let $t(P)$ be the treewidth of its cover graph.

Multiplicative lower-bound conjecture. Among cover-planar posets,

$$\dim(P)=\Omega(m(P)t(P)).$$

The paper proves the upper bound $\dim(P)\leq m(P)(4t(P)+6)$ and notes separate lower bounds of order $t(P)$ from wheels and of order $m(P)$ from Kelly posets. The conjecture asks whether the product lower bound is asymptotically attained; the source leaves this question open.