This question is motivated by If $(R:_{Q(R) } S)$ is non-zero, then does $(R:_{Q(R) } S)$ contain a non-zero-divisor? (and a now deleted comment on it).
Namely: Does there exist a one-dimensional Noetherian reduced local ring $(R,\mathfrak m)$ such that $(R:_{Q(R)} \overline R)$ is non-zero and consists of zero-divisors ?
Here, $\overline R$ denotes the integral closure of $R$ in the total ring of fractions $Q(R)$ of $R$.
Thoughts: Trivially, $R$ cannot be an integral domain. Also, note that if the $\mathfrak m$-adic completion of $R$ is reduced, then $\overline R$ is module finite over $R$ (https://stacks.math.columbia.edu/tag/032Y) , hence $(R:_{Q(R)} \overline R)$ contains a non-zero-divisor (namely, a common denominator of a finite generating set of $\overline R$ as $R$-module). Hence any example , if it exists, should have non-reduced completion, in particular $R$ must not be complete.