Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = {f\in K[x] \mid f(D) \subseteq D}$betheringofinteger-valuedpolynomialson$D$. Given any finite multiset $\{k_1,\dots ,k_n\}$ of integers greater than $1$, we construct a polynomial in $\mathrm{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\mathrm{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multiplicative monoid of $\mathrm{Int}(D)$ to a block monoid.