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

Let $R(n,m)$ be the set of positive integers $k$ for which some word of length $2n$ over a complementary alphabet of size $m$ has exactly $k$ valid rooted plane trees. Black–Drellich–Tymoczko's universality conjecture. For every positive integer $k$, there exist $n$ and $m$ such that

$$k\in R(n,m).$$

This asserts that every positive integer occurs as the number of valid plane trees for some choice of word length and complementary alphabet. The paper's abstract states that the two conjectures are resolved; unlike the first conjecture, the supplied context gives no explicit resolution or proof status for this claim.