The conformal-equivalence conjecture for GenRel without localized time symmetry

Let $GenRel$ be the first-order axiom system for general relativity, let $AxSymTime^-$ denote the localized time-symmetry axiom, and let Lorentzian manifolds be the standard models of general relativity. Conformal-equivalence conjecture. If we leave out $AxSymTime^-$ from $GenRel$, it will still capture Lorentzian manifolds up to conformal equivalence. This conjecture concerns the role of localized symmetry axioms in the first-order framework for general relativity: omitting them is expected to leave only the freedom of changing measurement units locally. The supplied text gives no resolution, so the conjecture remains open.