The local coset-covering function conjecture for Dedekind domains

Let $(R,\mathfrak{m})$ be a local Dedekind domain with finite residue field, and let $n_1,\ldots,n_l$ be positive integers. For the finite $R$-module

$$M=\bigoplus_{i=1}^l R/\mathfrak{m}^{n_i},$$

write $\phi(M)$ for its coset-covering function and $\phi'(\mathfrak{m},n_i)$ for the associated cyclic-module quantities. The local coset-covering function conjecture. One has

$$\phi\left(\bigoplus_{i=1}^l R/\mathfrak{m}^{n_i}\right)=\sum_i\phi'(\mathfrak{m},n_i).$$

This is stated as the local version equivalent to the preceding global conjecture, using the Chinese remainder theorem and earlier propositions and lemmas. It is not proved in the supplied text, so it remains open here.