Weak-field limit of Kaluza-Klein models with spherical compactification: problematic aspects

We investigate classical gravitational tests for the Kaluza-Klein model with spherical compactification of the internal two-dimensional space. In the case of the absence of a multidimensional bare cosmological constant, the only matter which corresponds to the proposed metric ansatz is a perfect fluid with the vacuum equation of state in the external space and the dust-like equation of state in the internal space. We perturb this background by a compact massive source with the dust-like equation of state in both external and internal spaces (e.g., a point-like mass), and obtain the metric coefficients in the weak-field approximation. It enables to calculate the parameterized post-Newtonian parameter $\gamma$. We demonstrate that $\gamma=1/3$ which strongly contradicts the observations.