Zeilberger–Reed conjecture on 3-connected planar linear normal lambda-terms

A planar linear normal $\lambda$-term is a linear normal $\lambda$-term whose syntactic diagram is planar; call it 3-connected when its syntactic diagram is 3-connected. For $n\geq 0$, consider such terms with $n+2$ variables. Zeilberger–Reed conjecture. The number of 3-connected planar linear normal $\lambda$-terms with $n+2$ variables is

$$\frac{2^n}{(n+1)(n+2)}\binom{2n+1}{n},$$

which is also the number of bipartite planar maps with $n$ edges. This conjecture proposes a precise enumerative correspondence between highly connected planar lambda-terms and bipartite planar maps; the supplied source gives no resolution status.