If $H=(V,E)$ is a hypergraph then we say that $C\subseteq V$ is a cutset if $C\cap e \neq \emptyset$ for all $e\in E$. We set $$\text{cut}(H) = \min\{|C|: C \text{ is a cutset of }H\}.$$ A subset $D\subseteq E$ is said to be a disjoint if the members of $M$ are mutually disjoint. Set $$\text{disj}(H) = \sup\{|D|: D\subseteq E\text{ is disjoint}\}.$$ An easy argument shows that $\text{disj}(H) \leq \text{cut}(H)$ for any hypergraph $H$.
Question. If $G = (V,E)$ is a simple, undirected graph with $\text{cut}(G) = \text{disj}(G)$, is there necessarily a disjoint set $D_0\subseteq E$ and a (minimal) cutset $C_0$ such that $|C_0\cap d| = 1$ for all $d \in D_0$?