Converse to the logical and probabilistic exclusivity implication

Let $A$ be a partial Boolean algebra. Say that $A$ satisfies Logical Exclusivity (LEP) when its commeasurability structure has the corresponding logical exclusivity property, and say that all states of $A$ satisfy Probabilistic Exclusivity (PEP) when probabilistic exclusivity holds for every state.

LEP--PEP converse conjecture. If PEP holds, then LEP holds:

$$\mathrm{PEP}\;\Rightarrow\;\mathrm{LEP}.$$

The paper establishes the implication from LEP to PEP and conjectures the converse, so the two notions would coincide if this conjecture were proved. No resolution is supplied in the source.