Representability of maxoids by positive discrete distributions

A maxoid is a conditional-independence structure arising from the maximum-weight trek systems of a weighted acyclic directed graph. A discrete distribution is positive when it assigns strictly positive probability to every configuration.

Representability conjecture. Every maxoid is representable as the conditional-independence structure of a positive discrete distribution.

The conjecture asks whether the representability obstruction found for regular Gaussian distributions disappears for positive discrete distributions. It is posed as an open problem and is described as a stronger conjecture motivated by the lack of apparent reasons that maxoids should fail to be discretely representable.