Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$

Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary relation $R$ on a set of size $n$.

I have verified the statement for $n \leq 4$ with a Mathematica code.I have tried to prove the statement by considering the digraph $\Gamma(R)$ associated to $R$. I tried to use this fact: If $(x,y) \in \bigcap \limits_{\ell=1}^n R^\ell$ then there is a directed walk in $\Gamma(R)$ from $x$ to $y$ of every integer length $\ell \leq n$. This implies that $\Gamma(R)$ contains a cycle. I would like to show that this also implies that there is a directed walk in $\Gamma(R)$ from $x$ to $y$ of length $n+1$.