The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{ \mathcal{ M}_{ 0,n}}$, in the Grothendieck ring of varieties encodes its Poincar\'e polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{ \mathcal{ M}_{ 0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{ rk}\, H^{ 2l}(\overline{ \mathcal{ M}_{ 0,n}})$. As a consequence, we prove that in the range $l \le \frac{ n}{ 10\log n}$, the ultra-log-concavity inequality \[\left(\frac{ \mathrm{ rk}\, H^{ 2(l-1)}(\overline{ \mathcal{ M}_{ 0,n}})}{ \binom{ n-3}{ l-1}}\right)^2 \ge \frac{ \mathrm{ rk}\, H^{ 2(l-2)}(\overline{ \mathcal{ M}_{ 0,n}})\mathrm{ rk}\, H^{ 2l}(\overline{ \mathcal{ M}_{ 0,n}})}{ \binom{ n-3}{ l-2}\binom{ n-3}{ l}} \] holds for $n$ sufficiently large.
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