Model of ordered plane with continuity and at least one line not isomorphic to $\mathbb{R}$.

Question

Background information

Among primitive notions in mamy axiomatic systems for geometry we have ternary betweenness relation. Let's look at the following axioms of strict one-dimensional betweenness:

1. If $B(abc)$, then $a\neq c$
2. If $B(abc)$, then $B(cba)$
3. If $B(abc)$, then $\neg B(bac)$
4. If $a,b,c$ are pairwise distinct, then $B(abc)\vee B(bac)\vee B(acb)$
5. If $B(abc)\wedge B(acd)$, then $B(bcd)$
6. If $B(abc)\wedge B(bcd)$, then $B(acd)$

In a sense they characterize linear order as the following theorem says

Thm. Assume the set $L$ has at least three elements and $B\subseteq L\times L\times L$ is a relation satisfying axioms 1-6. Then there exist exactly two binary relations $\prec$ such that $$B(abc) \iff \left(a\prec b\prec c \vee c\prec b\prec a\right)$$ and they are mutually inverse strict linear orders. Conversely, having linearly ordered set we can define the betweenness relation satisfying axioms 1-6.

The following theorem gives the sufficient condition for the linearly ordered set to be isomorphic to $(\mathbb{R},<)$

Thm. Assume $(L,\prec)$ is a linearly ordered set such that

1. There exist at least two distinct elements in $L$
2. There is no smallest and no greatest element in $L$
3. $\prec$ is countably dense
4. $\prec$ is Dedekind-complete

Then $(L,\prec)$ is isomorphic to $(\mathbb{R},<)$.

The asumption of countable density is important and cannot be replaced with just density (counterexample being the modification of Alexandroff line).

In geometry axioms 1-6 are usually present (some of them are actually derivable with aid of other axioms and two-dimensional Pasch's axiom). Another axioms may ensure the validity of conditions 1-4 and therefore any line is isomorphic to $\mathbb{R}$.

Question.

I'm looking for the model of two-dimensional Hilbert's axioms of connection and order with Dedekind's continuity in which there exists at least one line not isomorphic to $\mathbb{R}$. Note that I exclude congruence and parallel axioms.

For the betweenness relation restricted to any line axioms 1-6 will hold. Next, conditions 1 and 2 for the respective linear order on this line will hold as well as condition 4 which is implied by Dedekind continuity axiom. This order will be dense, too. The only thing which presumably doesn't follow from this particular set of axioms and may prevent the line from being isomorphic to $\mathbb{R}$, is countable density.

My guess is some line must be of the shape similar to the Alexandroff line if such model exists. Or maybe countable density can be proven although it can't be proven in one dimension ?

EDIT. My definition of "countably dense": An ordered set is said to be countably dense when there exists a countable set $F$ such that for any $a\prec b$ there exists $q\in F$ such that $a\prec q\prec b$.