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Background: The Efimov effect is a universal phenomenon in physics whereby three-body systems are stabilized via the interaction of an unbound two-body sub-systems. A hypothetical state in $^{12}\mathrm{C}$ at 7.458 MeV excitation energy, comprising of a loose structure of three $α$-particles in mutual two-body resonance, has been suggested in the literature to correspond to an Efimov state in nuclear physics. The existence of such a state has not been demonstrated experimentally. Method: Using the combined data sets from two recent experiments, one with the TexAT TPC to measure $α$-decay and the other with Gammasphere to measure $γ$-decay of states in $^{12}\mathrm{C}$ populated by $^{12}\mathrm{N}$ and $^{12}\mathrm{B}$ $β$-decay respectively, we achieve high sensitivity to states in close-proximity to the $α$-threshold in $^{12}\mathrm{C}$. Results: No evidence of a state at 7.458 MeV is seen in either data set. Using a likelihood method, the 95\% C.L. $γ$-decay branching ratio is determined as a function of the $β$-decay feeding strength relative to the Hoyle state. In parallel, calculations of the triple-alpha reaction rate show the inclusion of the Efimov corresponds to a large increase in the reaction rate around $5 \times 10^{7}$ K. Conclusion: From decay spectroscopy - at the 95\% C.L., the Efimov state cannot exist at 7.458 MeV with any $γ$-decay branching ratio unless the $β$-strength is less than 0.7\% of the Hoyle state. This limit is evaluated for a range of different excitation energies and the results are not favorable for existence of the hypothetical Efimov state in $^{12}\mathrm{C}$. Furthermore, the triple-alpha reaction rate with the inclusion of a state between 7.43 and 7.53 MeV exceeds the rate required for stars to undergo the red giant phase.