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A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \perp\!\!\!\perp Y]$, $[X \perp\!\!\!\perp Z \mid U]$, $[Y \perp\!\!\!\perp U \mid Z]$ and $[Z \perp\!\!\!\perp U \mid XY]$ but whose entropy vector violates the Ingleton inequality.
This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold.
