Existence of a dfs measure that is not finitely approximated

Let $T$ be a theory, let $\mathcal{U}$ be a monster model of $T$, and let $\mu\in\mathfrak{M}_x(\mathcal{U})$ be a Keisler measure. A measure is dfs if it is definable and finitely satisfiable. Existence conjecture. There is a theory $T$ and a Keisler measure $\mu\in\mathfrak{M}_x(\mathcal{U})$ such that $\mu$ is dfs but not finitely approximated.

The examples preceding this conjecture exhibit dfs measures that are not finitely approximated in related settings, while also noting that the relevant measure need not extend to a global dfs measure. The conjecture asks whether such a measure exists for some theory in general.