The left LBS-tree edge-activity conjecture

A left LBS tree is a local binary search tree whose root has no right child; that is, it is a planar rooted tree in which each vertex has at most a left child and a right child, with left child smaller and right child larger than the vertex. For $0\leq k\leq n-1$, compare the numbers of such trees having specified edge types.

Left LBS-tree edge-activity conjecture. For $0\leq k\leq n-1$, the number of left LBS trees with $k$ edges of the form $(i+1,i)$ is equal to the number of left LBS trees with $k$ edges of the form $(n,i)$.

This conjecture was motivated by the Athanasiadis question and by the correspondence between regions of the Linial arrangement, LBS trees, and NBC trees. The authors report computer evidence but leave the claimed equality as a conjecture.