  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-Schr\"oter 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.
