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It has been known for more than 40 years that there are posets with planarcover graphs and arbitrarily large dimension. Recently, Streib and Trotterproved that such posets must have large height. In fact, all knownconstructions of such posets have two large disjoint chains with all points inone chain incomparable with all points in the other. Gutowski and Krawczykconjectured that this feature is necessary. More formally, they conjecturedthat for every $k\geq 1$, there is a constant $d$ such that if $P$ is a posetwith a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then$\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discusspossibilities of generalizing the result by relaxing the condition that thecover graph is planar.

Dimension of posets with planar cover graphs excluding two long incomparable chains