  General relativity is a fully conservative theory, but there exist other
possible metric theories of gravity. We consider non-conservative ones with a
parameterized post-Newtonian (PPN) parameter, $\zeta_2$. A non-zero $\zeta_2$
induces a self-acceleration for the center of mass of an eccentric binary
pulsar system, which contributes to the second time derivative of the pulsar
spin frequency, $\ddot{\nu}$. In our work, using the method in Will (1992), we
provide an improved analysis with four well-timed, carefully-chosen binary
pulsars. In addition, we extend Will's method and derive $\zeta_2$'s effect on
the third time derivative of the spin frequency, $\dddot{\nu}$. For PSR
B1913+16, the constraint from $\dddot{\nu}$ is even tighter than that from
$\ddot{\nu}$. We combine multiple pulsars with Bayesian inference, and obtain
an upper limit, $\left|\zeta_{2}\right|<1.3\times10^{-5}$ at 95% confidence
level, assuming a flat prior in $\log_{10} \left| \zeta_{2}\right|$. It
improves the existing bound by a factor of three. Moreover, we propose an
analytical timing formalism for $\zeta_2$. Our simulated times of arrival with
simplified assumptions show binary pulsars' capability in limiting $\zeta_{2}$,
and useful clues are extracted for real data analysis in future. In particular,
we discover that for PSRs B1913+16 and J0737$-$3039A, $\dddot{\nu}$ can yield
more constraining limits than $\ddot{\nu}$.
