Layer-monotonicity conjecture for Richardson infection times

Let $G$) be a graph with distinguished vertex $x$, and let $H=G\times\{0,1\}$ have two copies of $G$, with corresponding vertices joined by an edge. Run Richardson's model on $H$ from the single initially infected vertex $(x,0)$, and let $T(z)$ denote the first infection time of $z\in H$. For each $y\in G$, compare the corresponding vertices $(y,0)$ and $(y,1)$. Layer-monotonicity conjecture. For every $y\in G$, $T(y,0)$ is stochastically smaller than $T(y,1)$. This would formalize the suspected monotonicity of infection times with distance from the initially infected layer and would imply sharper covering-time bounds for Richardson's model on the cube. It remains open in the stated generality.