Monotonicity conjecture for implied Poisson parameters

Fix a graph $G$ with vertices $a$ and $b$, and let $ab^{\mathbin{\square} k}$ denote the event that there are $k$ disjoint open paths between $a$ and $b$. For each $k$, define $\lambda_k$ to be the unique number satisfying

$$\mathbf{P}(ab^{\mathbin{\square} k})=\sum_{i=k}^{\infty}\frac{\lambda_k^i}{i!e^{\lambda_k}}.$$

The implied-Poisson-parameter conjecture. The sequence $\{\lambda_k\}$ is decreasing. This is proposed as a strengthening of the paper's earlier conjecture on the probabilities of disjoint connections; the source gives no proof or disproof.