This is a related question to an older one.

If $H_i = (V_i, E_i)$ are hypergraphs for $i=1,2$ then we say they are *isomorphic* if there is a bijection $f: V_1 \to V_2$ such that for $A \subseteq V_1$ we have $$A\in E_1 \text{ if and only if } f(A) \in E_2.$$
We say that $H=(\omega, E)$ is a *strongly complete regular linear* hypergraph on $\omega$ if

- $|e| = \aleph_0$ for all $e\in E$,
- $e_1\neq e_2\in E \implies |e_1\cap e_2| = 1$, and
- for all $m,n\in\omega$ there is $e\in E$ such that $\{m,n\}\subseteq e$.

**Question.** If $H_i = (\omega, E_i)$ are strongly complete regular linear hypergraphs for $i = 1,2$, are $H_1$ and $H_2$ necessarily isomorphic?