Graviton density-family independence conjecture for perturbative amplitudes

Let $\eta_{\mu\nu}$ be the Minkowski background metric and let $g$ be the spacetime metric. For $\omega\in[0,1]$, define the graviton density family by

$$\mathfrak{h}^{\{\omega\}}_{\mu\nu}:=\frac{1}{\varkappa}\left(\left(-\operatorname{Det}(g)\right)^{\omega/2}g_{\mu\nu}-\eta_{\mu\nu}\right).$$

It interpolates between the graviton field at $\omega=0$ and the graviton density at $\omega=1$.

Graviton density-family conjecture. The perturbative expansion with respect to $\mathfrak{h}^{\{\omega\}}_{\mu\nu}$ depends on $\omega$ only through the external-leg structure. Consequently, the corresponding Feynman integrals and Ward identities are equivalent for all $\omega\in[0,1]$, modulo transformations on the external graviton legs of the amplitudes.

The claim would make the choice of tensor-density weight irrelevant to internal perturbative dynamics, while allowing the external graviton-field parametrization to change. The source says that first propagator calculations suggest this statement and gives no resolution evidence.