  In the framework of the Parameterized Post-Newtonian (PPN) formalism, we
calculate the long-term Preferred Location (PL) effects, proportional to the
Whitehead parameter $\xi$, affecting all the Keplerian orbital elements of a
localized two-body system, apart from the semimajor axis $a$. They violate the
gravitational Local Position Invariance (LPI), fulfilled by General Relativity
(GR). We obtain preliminary bounds on $\xi$ by using the latest results in the
field of the Solar System planetary ephemerides. The non-detection of any
anomalous perihelion precession for Mars allows us to indirectly infer
$|\xi|\leq 5.8\times 10^{-6}$. \textcolor{black}{Such a bound is close to the
constraint, of the order of $10^{-6}$, expected from the future BepiColombo
mission to Mercury. As a complementary approach, the PL effects should be
explicitly included in the dynamical models fitted to planetary data sets to
estimate $\xi$ in a least-square fashion in a dedicated ephemerides orbit
solution.} The ratio of the anomalous perihelion precessions for Venus and
Jupiter, determined with the EPM2011 ephemerides at the $<3\sigma$ level, if
confirmed as genuine physical effects needing explanation by future studies,
rules out the hypothesis $\xi \neq 0$. A critical discussion of the $|\xi|
\lesssim 10^{-6}-10^{-7}$ upper bounds obtained in the literature from the
close alignment of the Sun's spin axis and the total angular momentum of the
Solar System is presented.
