  In this article, we give a short algebraic proof that all closed intervals in
a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter
group $W$ and any Coxeter element $\gamma\in W$. This means that if such an
interval has length $k$, then there exists a maximal chain of length $k$
consisting of left-modular elements, and there are precisely $k$ join- and $k$
meet-irreducible elements in this interval. Consequently every graded interval
in $\mathcal{C}_{\gamma}$ is distributive. This problem was open for any
Coxeter group that is not a Weyl group.
