Strict $2$-convexity of convex solutions to the quadratic Hessian equation

We prove that convex viscosity solutions to the quadratic Hessian inequality \begin{equation*} \sigma _2(D^2u) \geq 1 \end{equation*} are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for $\sigma = 1$, which were proven using different methods in recent works Guan-Qiu, McGonagle-Song-Yuan, Shankar-Yuan.