The set-theoretic Kaufmann-Clote question

Let $\mathsf{M}$ be the set theory obtained from $\mathsf{Z}\mathsf{F}$ by removing the collection scheme, restricting separation to ${\mathrm{\Delta }}_0$-formulae and adding an axiom asserting that every set is contained in a transitive set. Let ${\mathrm{\Pi }}_n\mathsf{\text{-Collection}}$ denote the restriction of the collection scheme to ${\mathrm{\Pi }}_n$-formulae. In this paper we prove that for $n\ge 1$, if $\mathcal{M}$ is a model of $\mathsf{M}+{\mathrm{\Pi }}_n\mathsf{\text{-Collection}}+\mathsf{V}=\mathsf{L}$ and $\mathcal{N}$ is a ${\mathrm{\Sigma }}_{n+1}$-elementary end extension of $\mathcal{M}$ that satisfies ${\mathrm{\Pi }}_{n-1}\mathsf{\text{-Colelction}}$ and that contains a new ordinal but no least new ordinal, then ${\mathrm{\Pi }}_{n+1}\mathsf{\text{-Collection}}$ holds in $\mathcal{M}$. This result is used to show that for $n\ge 1$, the minimum model of $\mathsf{M}+{\mathrm{\Pi }}_n\mathsf{\text{-Collection}}$ has no ${\mathrm{\Sigma }}_{n+1}$-elementary end extension that satisfies ${\mathrm{\Pi }}_{n-1}\mathsf{\text{-Collection}}$, providing a negative answer to the generalisation of a question posed by Kaufmann.