Heritability of Kőnig's Property from finite edge sets

A hypergraph $H=(V,E)$ is said to have Kőnig’s Property if there is a matching $M\subseteq E$ and $S\subseteq V$ such that $\mathrm{\mid }S\cap e\mathrm{\mid }=1$ for all $e\in M$, and $S$ is a vertex cover of $H$. Aharoni posed the question whether Kőnig’s Property is inheritable from finite subsets of $E$. We provide a negative answer and investigate similar questions for weaker properties.