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We seek signatures of the current experimental $^{12}$C$(α,γ)^{16}$O reaction rate probability distribution function in the pulsation periods of carbon-oxygen white dwarf models. We find that adiabatic g-modes trapped by the interior carbon-rich layer offer potentially useful signatures of this reaction rate probability distribution function. Probing the carbon-rich region is relevant because it forms during the evolution of low-mass stars under radiative helium burning conditions, mitigating the impact of convective mixing processes. We make direct quantitative connections between the pulsation periods of the identified trapped g-modes in variable WD models and the current experimental $^{12}$C$(α,γ)^{16}$O reaction rate probability distribution function. We find an average spread in relative period shifts of $ΔP/P \simeq \pm$ 2\% for the identified trapped g-modes over the $\pm$ 3$σ$ uncertainty in the $^{12}$C$(α,γ)^{16}$O reaction rate probability distribution function -- across the effective temperature range of observed DAV and DBV white dwarfs and for different white dwarf masses, helium shell masses, and hydrogen shell masses. The g-mode pulsation periods of observed white dwarfs are typically given to 6-7 significant figures of precision. This suggests that an astrophysical constraint on the $^{12}$C$(α,γ)^{16}$O reaction rate could, in principle, be extractable from the period spectrum of observed variable white dwarfs.