The rooted-subtree enumeration conjecture for maximally extendable transfer systems

Let $n\geqslant 0$. A transfer system in $T(n)$ is maximally extendable if it has the maximal extension property described in the surrounding definitions, and let $\mathfrak{A}_m$ denote the number of rooted subtrees in a rooted planar tree with $m$ nodes. Enumeration conjecture. The number of maximally extendable transfer systems in $T(n)$ is

$$\mathfrak{A}_{n+2}.$$

By the preceding correspondence, the same count is conjectured for maximally stationary transfer systems in $T(n)$ and in $L(n)$. The quantity $\mathfrak{A}_m$ has an explicit binomial formula and recurrence, and the claim is motivated by numerical experiments; no proof or resolution is supplied here.