It is a well-known theorem that well-orderings can not be characterized in $L_{\omega_1,\omega}$. In particular, if $\psi$ is an $L_{\omega_1,\omega}$-sentence in a vocabulary $\tau$ that contains a binary symbol $<$ and $<$ is a linear order in all models of $\psi$, then either $<$ is a well-ordering of order-type $\alpha<\omega_1$, or otherwise, $\psi$ has models where < is not well-ordered.

The proofs that I am aware of produce a model $N$ of size $\aleph_1$ such that $\mathbb{Q}$ embeds into $<^N$.

Is the following generalization of the above theorem true? Let $\psi$ be an $L_{\omega_1,\omega}$-sentence and $\psi$ has a model (of any size) where $<$ is not well-ordered. If $\psi$ as a model of size $\kappa$, then there is a model of size $\kappa$ where $<$ is not well-ordered.

My conjecture is "yes", but I could not find a proof in the literature. The main obstacle is that the methods used to produce the $\aleph_1$-sized model do not generalize to arbitrary $\kappa$.