Minimal elementary end extensions

Suppose that ${\mathcal M}$ is a model of PA and ${\mathcal N}$ is a countably generated elementary end extension of ${\mathcal M}$. Let ${\mathfrak X}$ be the set of subsets of M that are coded by ${\mathcal N}$. Then ${\mathcal M}$ has a minimal elementary end extension that codes exactly the same subsets of M that ${\mathcal N}$ does iff every set that is $Π_1^0$-definable in $({\mathcal M},{\mathfrak X})$ is the union of countably many sets that are $Σ_1^0$-definable.

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