Alphabet-independence conjecture for the sets R(n,m)

Let $R(n,m)$ denote the set of numbers of valid plane trees associated with primary sequences having $n$ edges over an alphabet with $m$ complementary letter pairs. The sets $R(n,m)$ are therefore indexed by the number of edges and the alphabet size.

Alphabet-independence conjecture. The set $R(n,m)$ depends only on $n$.

The conjecture is supported by computations for $R(n,2)$ with $n\leq 6$, for $R(n,3)$ and $R(n,4)$ with $n\leq 5$, and for $R(n,m)$ with $n\leq 4$ and $m\leq 10$; in all these cases, $R(n,m)$ agrees with $R(n,1)$.