Are strongly complete regular linear hypergraphs on $\omega$ isomorphic?

Question

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

1. $|e| = \aleph_0$ for all $e\in E$,
2. $e_1\neq e_2\in E \implies |e_1\cap e_2| = 1$, and
3. 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?

Answer 1

If $H_1= P^2(\mathbb{Q})$ (the projective plane defined on $\mathbb{Q}$ with the points as vertices and lines as hyper-edges), and $H_2$ the Moulton plane defined on $\mathbb{Q}$, then $H_1$ and $H_2$ are both strongly complete regular linear hypergraphs but they are not isomorphic.

The reason is that, if $H_1$ and $H_2$ are isomorphic, Desargues' theorem would hold on both of them or neither of them, but Desargues' theorem holds on $H_1$ and not $H_2$.