Converse to the state-extension theorem for logical exclusivity

Let $A$ be a partial Boolean algebra, let $\eta\colon A\longrightarrow A[\perp]$ be the canonical map to its logical exclusivity extension, and let $\nu$ be a state of $A$. A state satisfies Probabilistic Exclusivity (PEP) when it has the probabilistic exclusivity property.

State-extension conjecture. If state $\nu$ of $A$ satisfies PEP, then there is a state $\hat{\nu}$ of $A[\perp]$ such that

$$\nu=\hat{\nu}\circ\eta.$$

This is conjectured as the converse to the paper's theorem asserting the corresponding implication for states on the logical exclusivity extension. The source gives no resolution.