The lexicographic shellability conjecture for the electrical poset

Let $EP_n$ be the graded poset of electrical networks on $n$ boundary vertices, with rank given by the number of edges in a critical representative. Lexicographic shellability conjecture. $EP_n$ is lexicographically shellable, and hence Cohen–Macaulay, spherical, and Eulerian. This would give strong topological information about the order complex of $EP_n$. The text reports verification of the Eulerian property for $n\leq 7$, agreement of homology with a sphere for $n\leq 4$, and no known $L$-labeling for $n\geq 4$; the general conjecture remains open.