Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{{\mathcal{M}}_{0,n}}$

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 Poincaré 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 $\mathrm{r}\mathrm{k}{H}^{2l}(\overline{{\mathcal{M}}_{0,n}})$. As a consequence, we prove that in the range $l\le \frac{n}{10\log n}$, the ultra-log-concavity inequality $${\left(\frac{\mathrm{r}\mathrm{k}{H}^{2(l-1)}(\overline{{\mathcal{M}}_{0,n}})}{\left(\genfrac{}{}{0px}{}{n-3}{l-1}\right)}\right)}^2\ge \frac{\mathrm{r}\mathrm{k}{H}^{2(l-2)}(\overline{{\mathcal{M}}_{0,n}})\mathrm{r}\mathrm{k}{H}^{2l}(\overline{{\mathcal{M}}_{0,n}})}{\left(\genfrac{}{}{0px}{}{n-3}{l-2}\right)\left(\genfrac{}{}{0px}{}{n-3}{l}\right)}$$ holds for $n$ sufficiently large.