Say that a finitely axiomatizable first-order theory $T$ (in a finite language $\Sigma$) is **persistently finitely axiomatizable** iff the set $T_{\mathsf{w/o=}}$ of equality-free $T$-theorems is finitely axiomatizable in first-order logic without equality.

Building off of [this earlier question](https://mathoverflow.net/questions/419472/is-there-a-finitely-axiomatizable-class-of-structures-whose-equality-free-theory), I would like to ask:

> Is there a finitely axiomatizable *relational* theory which is not persistently finitely axiomatizable?

Here are the only two relevant facts I currently know:

 - One natural attempt for proving a negative answer is the following. Let $$I(x,y)\equiv\bigwedge_{R\in \Sigma} \forall \overline{u},\overline{v}(R(\overline{u},x,\overline{v})\leftrightarrow R(\overline{u},y,\overline{v})).$$ Then we might expect to have $\vdash A\leftrightarrow A_I$ for all first-order sentences $A$, where $A_I$ is gotten from $A$ by replacing "$s=t$" with "$I(s,t)$" throughout. However, this breaks down: consider e.g. $A\equiv \exists x,y(x=y\wedge \neg I(x,y))$. (This counterexample was [pointed out by Emil Jerabek](https://mathoverflow.net/questions/419472/is-there-a-finitely-axiomatizable-class-of-structures-whose-equality-free-theory#comment1085488_422396).)

 - If we allow function symbols we get a **positive** answer; this was observed by [Rodrigo Freire](https://mathoverflow.net/a/422396/8133), answering the above-linked original question.