Is this modified gravity field equation with a memory term consistent with GR in the solar system limit? [closed]

I'm exploring a toy model where gravity arises from the "memory" of spacetime. The idea is that spacetime has a rigidity and retains deformation caused by matter.

The field equation I propose is: $ G_{\mu\nu} = \frac{8\pi G}{c^4} \left[ T_{\mu\nu} + \lambda \, H_{\mu\nu} \right] $ where $ H_{\mu\nu}(x,t) = \int_{-\infty}^{t} K(t-t') \, T_{\mu\nu}(x,t') \, dt' $ and $ \lambda = \frac{G}{c^4 l_p^2}, \quad K(t-t') \sim \frac{c}{r_0} e^{-(t-t')/\tau} $ with $r_0 = GM/a_0$ and $a_0 \approx 1.2 \times 10^{-10} m/s^2$.

Motivation:

1. For $r \ll r_0$: $H_{\mu\nu} \to 0$ so we recover GR. This should pass solar system tests.
2. For $r \sim r_0$: $G_{eff} = G(1 + r_0/r)$. This gives flat galaxy rotation curves without dark matter.
3. For clusters: The memory term lags behind, so lensing can separate from gas, like in Bullet Cluster.

My specific questions:

1. Is this equation covariant? Does the integral break diffeomorphism invariance?
2. In the weak field, static limit, does this reduce to a Poisson equation with $G_{eff}$?
3. Are there known no-go theorems against non-local-in-time gravity terms like $H_{\mu\nu}$?

I know this is speculative. I'm not claiming it's correct, just asking if the math is self-consistent.

References: Inspired by ideas from Loop Quantum Gravity spin foams and MOND phenomenology.