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Assuming the best numerical value for the cosmic baryonic density and the existence of three neutrino flavors, standard big bang nucleosynthesis is a parameter-free model. It is important to assess if the observed primordial abundances can be reproduced by simulations. Numerous studies have shown that the simulations overpredict the primordial $^7$Li abundance by a factor of $\approx$ $3$ compared to the observations. The discrepancy may be caused by unknown systematics in $^7$Li observations, poorly understood depletion of lithium in stars, errors in thermonuclear rates that take part in the lithium and beryllium synthesis, or physics beyond the standard model. Here, we focus on the likelihood of a nuclear physics solution. The status of the key nuclear reaction rates is summarized. Big bang nucleosynthesis simulations are performed with the most recent reaction rates and the uncertainties of the predicted abundances are established using a Monte Carlo technique. Correlations between abundances and reaction rates are investigated based on the metric of mutual information. The rates of four reactions impact the primordial $^7$Li abundance: $^3$He($α$,$γ$)$^7$Be, d(p,$γ$)$^3$He, $^7$Be(d,p)2$α$, and $^7$Be(n,p)$^7$Li. We employ a genetic algorithm to search for simultaneous rate changes in these four reactions that may account for all observed primordial abundances. When the search is performed for reaction rate ranges that are much wider than recently reported uncertainties, no acceptable solutions are found. Based on the currently available evidence, we conclude that it is highly unlikely for the cosmological lithium problem to have a nuclear physics solution.