The ordinal-condition characterization of the finite-model subtheory of MLSP

Let $\psi$ be an $\textnormal{\textsf{MLSP}}$-formula, let $\mathrm{Vars}(\psi)$ be its set of variables, and let $M$ be a model of $\psi$. The subtheory $\widehat{\textnormal{\textsf{MLSP}}}$ consists of those $\textnormal{\textsf{MLSP}}$-formulae that have no infinite models. For a model $M$, write $\text{\sf rk}(Mx)$ for the rank of the set assigned to $x$. Ordinal-condition conjecture. If $\psi \in \widehat{\textnormal{\textsf{MLSP}}}$, then

$$\big\{ \text{\sf rk}(Mx) \mid x \in \mathrm{Vars}(\psi) \big\}$$

is an ordinal for every model $M$ of $\psi$. This asserts the necessity of the ordinal condition for membership in $\widehat{\textnormal{\sf MLSP}}$; together with the preceding implication from the ordinal condition to membership, it would characterize this finite-model subtheory.