  We design a direct test of the local position invariance (LPI) in the
post-Newtonian gravity, using the timing observation of the triple pulsar, PSR
J0337+1715. The test takes advantage of the large gravitational acceleration
exerted by the outer white dwarf to the inner neutron star -- white dwarf
binary. Using machine-precision three-body simulations and dedicated
Markov-chain Monte Carlo (MCMC) techniques with various sampling strategies and
noise realizations, we estimate that the Whitehead's parameter could have
already been limited to $|\xi| \lesssim 0.4$ (95\% CL), with the published
timing data spanning from January 2012 to May 2013. The constraint is still
orders of magnitude looser than the best limit, yet it is able to independently
falsify Whitehead's gravity theory where $\xi=1$. In addition, the new test is
immune to extra assumptions and involves full dynamics of a three-body system
with a strongly self-gravitating neutron star.
