Let $R$ be a binary relation on $[n]$. Let $k$ be the index of $R$, i.e., the smallest positive integer such that $R^k = R^{k+m}$ for some $m \geq 1$. Let $m$ be the period of $R$, i.e., the smallest positive integer satisfying this equation. Then the relations $R^k,R^{k+1},R^{k+2},\ldots ,R^{k+m-1}$ form a cyclic group.
In David Rosenblatt, "On the graphs of finite idempotent Boolean relation matrices", it is stated without proof that $\bigcup_{i=k}^{k+m-1} R^i$ is idempotent.
I can see that this union is transitive because it is the transitive closure of a generator of the cyclic group.
I cannot prove that the union is dense, i.e., if $(x,y)$ is in the union then there is a $z$ such that $(x,z)$ and $(z,y)$ are also in the union.
This is actually true in any (additively) idempotent semiring $S$. Let $r \in S$ be such that $r^k = r^{k+m}$. As you pointed out, $G = \{r^k, \ldots, r^{k+m} \}$ is a cyclic group with identity $e$. In particular, for every $n \geqslant 0$, $r^{k+n} \in G$. It follows that $$ \sum_{i=k}^{k+m-1} r^i = \sum_{x \in G} x $$ is idempotent since $$ \bigl(\sum_{x \in G} x\bigr)\bigl(\sum_{y \in G} y\bigr) = \sum_{x \in G}xe + \sum_{x \in G}\sum_{y \in G - \{e\}} xy = \sum_{x \in G} x $$