Full Reeh-Schlieder conjecture for the free Dirac field

Let $M$ be a globally hyperbolic spin spacetime with a non-compact Cauchy surface. Let ${\mathbf{B}}$ denote the causal locally covariant theory used for the $C^*$-algebraic approach, and let ${\mathcal{T}}_M$ be its relevant state space. For each bounded cc-region $V\subset M$, let ${\mathcal{R}}_V$ denote the associated local von Neumann algebra. Full Reeh-Schlieder conjecture. There is a state $\omega\in{\mathscr{T}}_M$ on ${\mathcal{B}}_M$ such that $\mathcal{H}_{\omega}$ contains a dense $G_{\delta}$ set $\mathcal{G}$ of vectors which define full Reeh-Schlieder states. For all bounded cc-regions $V\subset M$ with non-zero causal complement, each vector $\psi\in\mathcal{G}$ is cyclic and separating for ${\mathcal{R}}_V$. This is proposed as a $C^*$-algebraic route to full Reeh-Schlieder states for the free Dirac field, since the density theorem used earlier requires causality; the source states that this would require working with ${\mathbf{B}}$ rather than $\overline{{\mathbf{F}}}^0$.