Conjecture on mixing models for Boolean-satisfiable infinitary theories

Let $\mathrm{L}$ be a relational $\omega$-signature. A theory in $\mathrm{L}_{\infty\infty}$ is Boolean satisfiable if it has a Boolean-valued model satisfying the theory. A Boolean-valued model has the mixing property when it satisfies the mixing condition for Boolean-valued semantics. Mixing-model conjecture. There are Boolean-satisfiable $\mathrm{L}_{\infty\infty}$-theories which do not have a Boolean-valued model with the mixing property. This conjecture proposes a limitation on strengthening Boolean completeness for $\mathrm{L}_{\infty\infty}$; the source provides no evidence that it has been resolved.