Using a known recursive formula for the Grothendieck classes of the moduli spaces \(\overline{{\mathcal {M}}}_{0,n}\) , we prove that they satisfy an asymptotic form of ultra-log-concavity as polynomials in the Lefschetz class. We also observe that these polynomials are \(\gamma \) -positive. Both properties, along with numerical evidence, support the conjecture that these polynomials only have real zeros. This conjecture may be viewed as a particular case of a possible extension of a conjecture of Ferroni-Schröter and Huh on Hilbert series of Chow rings of matroids. We prove asymptotic ultra-log-concavity by studying differential equations obtained from the recursion, whose solutions are the generating functions of the individual Betti numbers of \(\overline{{\mathcal {M}}}_{0,n}\) . We obtain a rather complete description of these generating functions, determining their asymptotic behavior; their dominant term is controlled by the coefficients of the Lambert W function. The \(\gamma \) -positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving \(\gamma \) -positivity for the Hilbert series of the Chow ring of matroids.
