general relativity - Non-local gravitational energy tensor

The well-known derivation of the Landau-Lifshitz gravitational energy pseudotensor, relies on several requirements:

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  1) that it be constructed entirely from the metric tensor

  
  



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  2) that it be index symmetric so that it conserves angular momentum

  
  

• 3) when added to the stress–energy tensor of matter, $T^{\mu \nu}$, its total 4-divergence vanishes (this is required of any conserved current) so that we have a conserved expression for the total stress–energy–momentum.

• 4) that it vanish locally in an inertial frame of reference (which requires that it only contains first and not second or higher derivatives of the metric). This is because the equivalence principle requires that the gravitational force field, the Christoffel symbols, vanish locally in some frame. If gravitational energy is a function of its force field, as is usual for other forces, then the associated gravitational pseudotensor should also vanish locally.

The only quantity that satisfies all the above turns out not to be a tensor. My question is:

Is there a consistent derivation that doesn't assume requirement 3) but that still manages to be a well-behaved tensor?

I suspect that removing this requirement leaves too many possible degrees of freedom