Is Einstein Equivalence principle a consequence of weak equivalence principle + covariance principle? I have been doing some thinking about the Einstein Equivalence Principle (EEP) and its formulation, namely:

The outcome of any local non-gravitational experiment in a freely falling laboratory isindependent of the velocity of the laboratory and its location in spacetime.

After some time it came to mind the following idea: isn't this formulation just a consequence of the weak equivalence principle plus the covariance principle?

A consequence of the WEP because a freely falling laboratory is equivalent to one in the absence of a gravitational field. But also a consequence of the covariance principle because in the absence of gravity the outcome of the experiment is velocity and position independent.
Any insight is welcome!
 A: The idea that the Einstein equivalence principle should follow from the weak equivalence principle is known as the Schiff's conjecture. In its original formulation it states that:

Every theory of gravity that satisfies the WEP and is relativistic necessarily satisfies the EEP, and is consequently a metric theory of gravity.

It is generally believed that rigorous proof of this conjecture is impossible, since it would necessitate understanding of all possible gravitational theories satisfying the WEP (including those not yet invented), but plausibility arguments in support of the conjecture have been put forward as well as proofs under constraints on possible gravitational theories and non-gravitational interactions (see Lightman & Lee, 1972 as an example).
At the same time there are known counterexamples to this conjecture (and thus examples of models satisfying the WEP but violating the EEP), such as those constructed from Einstein's general relativity with matter that couples non-minimally to the gravitational field (Example 1,  Example 2), but rather than completely invalidating the conjecture those examples suggest the necessity of fine-tuning its precise statement.
For more information on the topic see review:

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*Will, C. M. (2014). The confrontation between general relativity and experiment. Living reviews in relativity, 17(1), 4, doi:10.12942/lrr-2014-4.

