Let $H=(V,E)$ be a hypergraph. We say that $I\subseteq V$ is an independent set if $e\not\subseteq I$ for all $e\in E$.

We say that $H$ is tameable if every independent set is contained in a maximal independent set. Every graph is tameable and, more generally, so is every hypergraph with finite edges.

There are easy examples of non-tameable hypergraphs, and I use this one given by user @bof in the comment section of this answer: Let $H=(\omega,[\omega]^\omega)$, where $[\omega]^\omega$ denotes the collection of infinite subsets of $\omega$. (The only independent subsets of this graph are the finite sets, and there is no maximal finite set.)

If $(P,\leq)$ is a poset, then with $\text{Max}(P)$ we denote the collection of maximal elements of $P$. (Note that $\text{Max}(\omega) = \varnothing$, for instance.)

Given a hypergraph $H=(V,E)$ we let $$\text{Tame}(H) = \{E'\subseteq E: (V, E') \text{ is tameable}\}.$$

Question. Given a hypergraph $H=(V,E)$ with $V\neq\varnothing\neq E$ and $\varnothing\notin E$, do we necessarily have $\text{Max}(\text{Tame}(H)) \neq \varnothing$?