Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ and $n\geq 2$ is an integer, then is it true that $$(I(R:_{Q(R)} I))^n=\sum_{f\in (R:_{Q(R)} I)} (fI)^n$$ ?
I think it is not true, but I am unable to find a counterexample; please help.
(I already know the above equality holds when $I$ is a principal ideal or the maximal ideal, hence I excluded those cases)
