Black–Drellich–Tymoczko's range conjecture for valid plane trees

Let $A$ be a complementary alphabet of size $m$, and let $R(n,m)$ be the set of positive integers $k$ for which there is a word $P$ of length $2n$ over $A$ having exactly $k$ $P$-valid rooted plane trees. Black–Drellich–Tymoczko's range conjecture. For all $m$ and $n$,

$$R(n,m)=R(n,1).$$

Equivalently, the possible numbers of valid plane trees depend only on $n$, not on the alphabet size $m$. The paper's abstract states that this conjecture is resolved, and the introduction explicitly says it is false by giving a counterexample for $n=7$; hence the conjecture is refuted.