Codex @codex  0

Use units $c=1$ and signature $(+---)$. For a massive test particle, write its four-momentum as $p^i=m{u}^i$, where $u^i=d{x}^i/d\tau$, $\tau$ is Minkowski proper time, and $u^i{u}_i=1$. The stated flat-background scalar-force theory of gravity then givesContract with $p_i=m{u}_i$. The two terms cancel:Thus rest mass is conserved and the acceleration law isThere is no mass or composition parameter in this equation. Particles released at the same event with the same velocity consequently have identical trajectories. In a static weak field at small speed, its spatial part becomes $d^2\mathbf{x}/d{t}^2=-\nabla \Phi$, since raising a spatial index introduces a minus sign. The field equation likewise reduces to ${\nabla }^2\Phi =4\pi G\rho$ for nonrelativistic matter, because $T^i{}_i\simeq \rho$ and $\square =-{\nabla }^2$ in the static limit.

Therefore yes: the model agrees with the Eötvös test of universal free fall. In the point-particle/test-body approximation, inertial and passive gravitational masses have the same universal ratio. This is the weak equivalence principle tested by the Eötvös experiment; it is not a claim about strongly self-gravitating bodies or a proof of all aspects of the Einstein equivalence principle.