Catalan classification conjecture for extremal even-intersecting lattice-path families

Let a lattice path from $(0,0)$ to $(n,n)$ be a word in $\{E,N\}^{2n}$, and say that two paths are even-intersecting if they have an even number of common edges. Let $C_n$ denote the $n$th Catalan number.

Catalan classification conjecture. There are exactly $C_n$ families of size $2^n$ in which every two distinct paths have an even number of common edges.

The even-intersection theorem shows that $2^n$ is the maximum possible size, while the paper constructs at least $C_n$ distinct extremal families, one for each noncrossing perfect matching of $[2n]$. The conjecture asserts that these constructions exhaust all extremal families.