I am referring to the very useful K. Conrad's notes on the conductor ideal of an order in a Dedekind domain: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf

$\DeclareMathOperator\Cl{Cl}$On page 13 it is claimed that every ideal class of the order contains a representative coprime with the conductor, and this would be crucially needed in a paper I am writing. Unfortunately, the proof is omitted (Theorem 5.2). By Lemma 5.1, the statement is valid in the Dedekind domain, and the document suggests that this could be extended to the order by some work. For sure, one of the classes in the preimage via the extension map from $\Cl(\mathcal{O})$ to $\Cl(\mathcal O_{K})$ of an element of $\Cl(\mathcal{O}_{K})$ has the property, but I could not prove why this should also occur with the remaining ones.

Has anyone succeeded in completing this proof?

notprove it. I edited accordingly. I hope that this was correct. $\endgroup$