Interior $W^{2,p}$ regularity for the sigma-$2$ equation

We prove that continuous $2$-convex solutions of the ${\sigma }_2$ equation with a density bounded above and below by positive constants belong to $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}$ in every dimension. If, in addition, the density is continuous, the solutions belong to $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,p}$ for every $1<p<\mathrm{\infty }$.