Charleston and Steel's recurrence conjecture for ancestral sequence reconstruction

Let $T_k$) be a fully bifurcating phylogenetic tree of height $k$, let $R$ be an alphabet with $|R|=r\geq 2$, and let $f_{k,r}$ denote the minimum number of leaves assigned a fixed state $a\in R$ such that the Fitch set at the root is $\{a\}$. For positive integers $p$, the proposed recurrence is

Charleston and Steel's conjecture. For $T_k$ and $r\geq 2$,

$$f_{k,r}=\begin{cases} f_{k-p,r}+f_{k-p-1,r} & \text{when } r=2p,\ 2\cdot f_{k-p,r} & \text{when } r=2p-1, \end{cases}$$

where $p\in\mathbb{N}_{\geq 1}$ in the first case and $p\in\mathbb{N}_{\geq 2}$ in the second case.

The conjecture generalizes the known binary-alphabet result that $f_{k,2}$ equals the $(k+1)$th Fibonacci number. The source presents this as the general case proposed by M. Steel and M. Charleston; its resolution is not established by the supplied context.