Throughout, all structures are finite.
Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary iff it is the class of finite models of a sentence in the expansion of first-order logic by Hilbert's $\varepsilon$-operator which – on finite structures – do not depend on the choice of $\varepsilon$-operator used.
Separately, for $\varphi$ a first-order sentence in a language $\Sigma$, say that a class of structures $\mathbb{K}$ in a language $\Pi$ (for simplicity disjoint from $\Sigma$) is $\mathsf{FOL}_\varphi^\text{inv}$-elementary iff there is some $\Sigma\sqcup\Pi$-sentence $\theta$ such that the following are equivalent:
$\mathcal{M}\in\mathbb{K}$.
$\mathcal{M}$ has some expansion satisfying $\theta\wedge\varphi$.
Every expansion of $\mathcal{M}$ satisfying $\varphi$ also satisfies $\theta$.
Note that a priori no $\mathsf{FOL}_\varphi^\text{inv}$-elementarity notion need coincide exactly with $\mathsf{FOL}_\varepsilon^\text{inv}$-elementarity, since the $\varepsilon$-operator is fundamentally a second-order object. For instance, the question of whether $\mathsf{FOL}_\varepsilon^\text{inv}$ coincides with $\mathsf{FOL}_\lambda^\text{inv}$, where $\lambda$ says that $<$ is a linear order of the domain, is stated as an open problem in Otto's Epsilon logic is more expressive tha first-order logic over finite structures (and my understanding is that this is still open).
My question is whether any first-order sentence is known to capture $\varepsilon$:
Is there some $\varphi$ such that the $\mathsf{FOL}_\varphi^\text{inv}$-elementary classes are exactly the $\mathsf{FOL}_\varepsilon^\text{inv}$-elementary classes?
