Paracausal equivalence criterion for globally hyperbolic spacetimes

Let $t$ and $t'$ be Cauchy temporal functions for globally hyperbolic spacetimes $(\mathsf{M},g)$ and $(\mathsf{M},g')$. Denote by $\langle\cdot,\cdot\rangle$ the natural pairing between $\mathsf{T}^*\mathsf{M}$ and $\mathsf{T}\mathsf{M}$. The vector fields $\partial_t$ and $\partial_{t'}$ are dual to $dt$ and $dt'$ with respect to $g$ and $g'$, respectively.

Paracausal equivalence conjecture.

$$g\simeq g'\qquad \text{if and only if} \qquad \langle \partial_t,dt'\rangle>0 \; \text{ and }\; \langle \partial_{t'},dt\rangle>0.$$

The conjecture proposes a criterion for paracausal relatedness and would clarify when globally hyperbolic metrics determine algebraically equivalent free quantum field theories. The source explicitly leaves the claim to be proved or disproved, so its resolution is not established here.