  We establish new quantitative Hessian integrability estimates for viscosity
supersolutions of fully nonlinear elliptic operators. As a corollary, we show
that the optimal Hessian power integrability $\varepsilon =
\varepsilon(\lambda, \Lambda, n)$ in the celebrated $W^{2,
\varepsilon}$-regularity estimate satisfies $$\frac{ \left (1+
\frac{2}{3}\left(1- \frac{\lambda}{\Lambda} \right )\right )^{n-1}}{\ln n^4}
\cdot \left( \frac{\lambda}{\Lambda} \right) ^{n-1} \le \varepsilon \le
\frac{n\lambda}{(n-1)\Lambda +\lambda}, $$ where $n\ge 3$ is the dimension and
$0< \lambda < \Lambda$ are the ellipticity constants. In particular, $\left(
\frac{\Lambda}{\lambda} \right) ^{n-1} \varepsilon(\lambda, \Lambda, n)$
blows-up, as $n\to\infty$; previous results yielded fast decay of such a
quantity. The upper estimate improves the one obtained by Armstrong, Silvestre,
and Smart in arXiv:1103.3677
