Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following?

- $<$ is a linear order on a definable subset;
- $\phi$ is $\aleph_1$-categorical;
- in the unique model of $\phi$ of size $\aleph_1$, $<$ is also of size $\aleph_1$.

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