(We work in $\mathsf{ZFC+V=L}$.)

Define a **plausible theory** to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ which is $\omega_1$-c.e. and $\omega_1$-finitely satisfiable; that is, $T$ is $\Sigma_1$-definable over $L_{\omega_1}$ (with parameters) and every countable subtheory of $T$ (= every subtheory which is an element of $L_{\omega_1}$) is satisfiable. Note that since $L_{\omega_1}$ is uncountable, Barwise compactness does not apply and plausible theories need **not** be satisfiable. That said, every plausible theory is "generically" satisfiable: they all become satisfiable after forcing with $Col(\omega,\omega_1)$, since in the generic extension we can apply Barwise compactness. 

I'm interested in measuring the difficulty of "satisfiabilizing" a plausible theory via forcing. Specifically, set $T_0\trianglelefteq T_1$ iff $T_0$ is satisfiable in every generic extension in which $T_1$ is satisfiable. *(Note that by absoluteness of $\mathcal{L}_{\omega_1,\omega}$-semantics we don't have to worry about theories becoming unsatisfiable.)* As usual, this preorder induces an equivalence relation $\approx$ and a corresponding poset $\mathcal{Plaus}$ of **plausibility degrees**.

The already-satisfiable plausible theories constitute the least degree ${\bf 0}$, and there is a greatest degree ${\bf 1}$ as well: we can whip up a plausible theory $T_{max}$ describing a structure which $(i)$ is a countable linear order and $(ii)$ has each countable ordinal as an initial segment, and in order to make $T_{max}$ satisfiable we have to make $\omega_1$ countable. There are also intermediate degrees. For example, we can whip up a plausible theory describing $(\omega;<)$ equipped with a unary predicate which does not correspond to any constructible real, which becomes satisfiable exactly when we add a non-constructible real. This trick can be extended to "simple" forcing notions to get a lot more examples - e.g. there are plausible theories corresponding to the existence of a sufficiently Cohen generic real and to the existence of a sufficiently Sacks generic real, the pair of which show that $\trianglelefteq$ is not total and the latter of which shows that $\mathcal{Plaus}$ has atoms (= minimal nonzero degrees). It's also not hard to show that there are exactly $\omega_1$-many plausibility degrees.

I'm generally interested in the "shape" of $\mathcal{Plaus}$, but in particular the following question seems particularly natural:

> Does $\mathcal{Plaus}$ have *coatoms*?

That is, is there a plausibility degree ${\bf d}\triangleleft{\bf 1}$ with no ${\bf e}\in ({\bf d},{\bf 1})$? I strongly suspect that the answer is negative, but I don't see how to prove that.