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We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky's stable finiteness conjecture. Namely, the latter holds true if and only if for every finitely generated abelian group $A$ and every finitely generated Hopfian group $Γ$ the wreath product $A \wr Γ$ is Hopfian. In fact, we characterize precisely when $A \wr Γ$ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over $\mathbb{F}_p[Γ]$, for every prime $p$ occurring as the order of some element in $A$. A tool in our arguments is the fact that fields of positive characteristic locally embed into matrix algebras over $\mathbb{F}_p$ thus reducing the stable finiteness conjecture to the case of $\mathbb{F}_p$. A further application of this result shows that the validity of Kaplansky's stable finiteness conjecture is equivalent to a version of Gottschalk's surjunctivity conjecture for additive cellular automata.