Polygonality conjecture for framing posets of strongly planar DAGs

Let $G$ be a strongly planar directed acyclic graph, and let $\mathcal L$ be its framing poset. If $x$ and $y$ cover a common element $z$ and there exists a least upper bound $x\lor y$ for $x$ and $y$, then the interval $[z,x\lor y]$ is a square or a hexagon; dually, the analogous statement holds for lower covers and greatest lower bounds. Polygonality conjecture. Framing posets of strongly planar directed acyclic graphs display polygonality properties in which the relevant polygons are squares and hexagons. This conjecture is motivated by the polygonality of framing lattices, where squares, pentagons, and hexagons occur; the source presents the framing-poset assertion as an observed property and does not establish it generally.