State-dependent contextuality implies state-independent AvN in a partial closure

Let $\mathcal{M}$ be a measurement cover and let $\psi$ be a state. Say that $\mathcal{M}$ realizes a contextual empirical model for $\psi$ if the empirical model obtained from these measurements and this state is contextual. A partial closure is the closure operation on the measurement scenario considered in the paper.

Partial-closure AvN conjecture. Any measurement cover $\mathcal{M}$ that realizes a contextual empirical model for some state $\psi$ is state-independently $\mathrm{AvN}$ in a partial closure.

This conjecture proposes that state-dependent contextuality becomes state-independent all-versus-nothing contextuality after passing to a partial closure. The paper illustrates this relationship through examples including Mermin's square, CHSH scenarios, and $XZ$ scenarios, but the general assertion is presented without a proof.