The coset-covering function conjecture for finite modules over Dedekind domains

Let $R$ be a Dedekind domain satisfying the assumptions above, let $\mathfrak{m}_i$ be maximal ideals of $R$, and let $n_i$ be positive integers. Let $M$ be the finite direct sum

$$M=\bigoplus_i R/\mathfrak{m}_i^{n_i}.$$

Write $\phi(M)$ for the coset-covering function of $M$, and let $\phi'(\mathfrak{m}_i,n_i)$ denote the corresponding local quantities. The coset-covering function conjecture. One has

$$\phi(M)=\sum_i\phi'(\mathfrak{m}_i,n_i).$$

The statement reduces, via the Chinese remainder theorem and the preceding local results, to the corresponding assertion for finite direct sums of cyclic modules over a local Dedekind domain with finite residue field. The local assertion is not proved in the supplied text, so its resolution remains open here.