The graph dissimilarity metric conjecture

Let $G_1$ and $G_2$ be graphs with the same number of vertices, and let $\Delta$ be the dissimilarity measure obtained from the collections of horizontal homologies partitioned by the number of black-coloured vertices. Metric conjecture. The dissimilarity $\Delta$ is a metric on the set of graphs, meaning that

$$\Delta(G_1,G_2)>0$$

whenever $G_1\neq G_2$. The dissimilarity is already known to be a pseudometric for graphs with a fixed number of vertices; the conjecture asserts that it separates distinct graphs.