The graphical biclosed ornamentation lattice conjecture

For a finite undirected graph $G$, let $\mathsf{B}$ be the graphical pointed building set of $G$, and let $\operatorname{Bicl}(\mathsf{B})$ denote the subposet of biclosed ornamentations in $\mathcal{O}(\mathsf{B})$. Graphical biclosed ornamentation lattice conjecture. The poset

$$\operatorname{Bicl}(\mathsf{B})$$

is a lattice. The claim is motivated by several natural examples in which this poset is a lattice, while the paper notes that it is not a lattice for arbitrary pointed building sets; whether the graphical case always has this property remains open.