Matúš's rational-point conjecture for discrete conditional independence models

Let $\mathcal{C}$ be a discrete conditional independence model, let $J_{\mathcal{C}}$ be its conditional-independence ideal, and let $V(J_{\mathcal{C}})$ denote the associated variety of distributions. A distribution $p$ has rational joint probabilities when every joint probability of $p$ is rational.

Matúš's rational-point conjecture. For any discrete conditional independence model $\mathcal{C}$, there exists a distribution

$$p \in V(J_{\mathcal{C}})$$

such that all joint probabilities of $p$ are rational.

The conjecture concerns rational points on varieties defined by conditional independence constraints and is presented as a connection with an earlier conjecture by Matúš. Its resolution is not specified in the supplied text.