Equivalence of Proca Hadamard-state definitions on globally hyperbolic spacetimes

Let a globally hyperbolic spacetime be given, and consider the two Proca Hadamard-state conditions referred to as Definition~ and Definition~. Equivalence conjecture. The two definitions are equivalent on any globally hyperbolic spacetime. The paper establishes this equivalence when the spacetime has compact Cauchy surfaces and expects the result to extend to all globally hyperbolic spacetimes.