Valuation symmetry for incompatible quantum propositions

Let $|\Psi\rangle$ be a quantum state, let $\hat{P}$ be the operator associated with an experimental proposition, and let $v$ be the function assigning images to truth values. Valuation symmetry conjecture. Under $v$, the truth of

$$|\Psi\rangle\notin\operatorname{ran}(\hat{P})\sqcap|\Psi\rangle\notin\operatorname{ker}(\hat{P})$$

has the same image as the truth of $|\Psi\rangle\in\operatorname{ran}(\hat{P})$ or the truth of $|\Psi\rangle\in\operatorname{ker}(\hat{P})$. The source introduces this as an additional assumption intended to make the equal-cost conjecture possible without implying the quantum parallelism efficiency conjecture; no resolution is supplied.