Ergodic decomposition of invariant measures on structure spaces

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be a cardinal. Let $\mu$ be an invariant measure on $\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)$. Ergodic-decomposition conjecture. The measure $\mu$ is a mixture of ergodic invariant measures on $\operatorname{Str}_{\kappa}[\operatorname{\mathscr{L}}](X)$. The paper recalls this decomposition when $X$, $\operatorname{\mathscr{L}}$, and $\kappa$ are countable and conjectures that it remains true when at least one of them is uncountable.