Matúš's rational distribution conjecture for conditional independence models

Let ${\mathcal C}$ be a discrete conditional independence model, and let $J_{\mathcal C}$ denote its conditional independence ideal. A distribution is a point in the zero set of $J_{\mathcal C}$ when it satisfies all polynomial constraints in that ideal.

Matúš's conjecture. There exists a distribution $p$ in the zero set of $J_{\mathcal C}$ such that all joint probabilities of $p$ are rational.

This conjecture asks whether every discrete conditional independence model has a rational point in its associated algebraic variety. The source attributes it to Matúš; its resolution is not established by the supplied text.