Finitarity and definability of the closure operator

Let $\operatorname{cl}$ be the invariant closure operator induced by the independence relation mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}} on the monster structure $\mathfrak C$.

Finitarity and definability conjecture. $\operatorname{cl}$ is always finitary. Moreover, $\operatorname{cl}$ is definable: for every small $A\subseteq\mathfrak C$, the set $\operatorname{cl}(A)$ is ord-definable over $A$.

The preceding proposition establishes finitarity when the independence relation is superior, but the general finitarity and definability assertions are left as conjectural properties of the closure operator.