Monotonicity conjecture for the number of galled networks

Let $\mathrm{GN}_{n,k}$ denote the number of galled networks with parameters $n$ and $k$. Monotonicity conjecture. The sequence $\mathrm{GN}_{n,k}$ is increasing for $0\leq k\leq n$ and decreasing for $n\leq k\leq 2n-2$. This conjecture describes the expected monotonicity of the number of galled networks as the parameter $k$ varies; the paper states that the claims are not needed for its subsequent results and that proofs might appear elsewhere.