Monotonicity conjecture for stream-structure graphs

Let $F^t$ be a semi-flow, and let $\operatorname{Down}$ and $\operatorname{Down}^*$ be stream structures with graphs $\Gamma$ and $\Gamma^*$, respectively. Assume every node of $\operatorname{Down}$ is a subset of a node of $\operatorname{Down}^*$. Let $N_1,N_2$ be nodes of $\operatorname{Down}$ and let $N_1^*,N_2^*$ be nodes of $\operatorname{Down}^*$ such that $N_1 asubseteq N_1^*$ and $N_2 asubseteq N_2^*$.

Monotonicity conjecture. If $N_1\thicksim_S N_2$ in $\operatorname{Down}$, then $N_1^*\thicksim_S N_2^*$ in $\operatorname{Down}^*$; equivalently, $\Gamma$ maps into $\Gamma^*$.

This conjecture asserts that enlarging the nodes preserves the graph relation. The supplied text gives no indication that it has been proved or refuted.