Given a set of marginals, what is the largest support of a distribution satisfying these?

Given a random variable $$X$$ with support over $$\{0,1\}^I$$, we can define the marginal distribution on the bits indexed by $$A \subseteq I$$ by $$Pr(X_A = x_A) = \sum_{x \in \{0,1\}^{I - A}} Pr(X = x \cup x_A)$$. Given we know some marginal distribution over our full distribution, if that marginal has any $$x_A$$ having zero probability, it is clear that the full distribution must have all strings with $$x_A$$ as part having zero probability. We can define the extended support of any marginal as all strings which aren't set to zero probability by that marginal. Is it the case that, given a number of marginal constraints, if there exists a distribution which satisfies these constraints then there exists one which has support on the intersection of all the extended supports of the marginals?