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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, $ζ_2$. A non-zero $ζ_2$ 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, $\ddotν$. 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 $ζ_2$'s effect on the third time derivative of the spin frequency, $\dddotν$. For PSR B1913+16, the constraint from $\dddotν$ is even tighter than that from $\ddotν$. We combine multiple pulsars with Bayesian inference, and obtain an upper limit, $\left|ζ_{2}\right|<1.3\times10^{-5}$ at 95% confidence level, assuming a flat prior in $\log_{10} \left| ζ_{2}\right|$. It improves the existing bound by a factor of three. Moreover, we propose an analytical timing formalism for $ζ_2$. Our simulated times of arrival with simplified assumptions show binary pulsars' capability in limiting $ζ_{2}$, and useful clues are extracted for real data analysis in future. In particular, we discover that for PSRs B1913+16 and J0737$-$3039A, $\dddotν$ can yield more constraining limits than $\ddotν$.