Monotonicity in the number of walkers

Let $G$ be a vertex-transitive graph with stationary distribution $\pi$. For $k\;\geqslant\;2$, let $\mathrm{SC}(G,k)$ be the social connectivity time when $k$ walkers independently start from vertices sampled according to $\pi$. Monotonicity in the number of walkers. The expectation $\mathbb{E}[\mathrm{SC}(G,k)]$ is monotonically non-increasing in $k$ for $k\;\geqslant\;2$. Thus, adding walkers should not increase the expected social connectivity time on a vertex-transitive graph.