The quadratic-form conjecture for the $n-2$ Hessian equation

Let $\bar\Gamma_k$ denote the Gårding cone for the $k$-th elementary symmetric polynomial $4\sigma_k\4$, and write $4\sigma_k^{jj}(\kappa)=\partial \sigma_k/\partial \kappa_j4$ and $4\sigma_k^{ii,jj}(\kappa)=\partial^2\sigma_k/(\partial \kappa_i\partial \kappa_j)4$. Suppose that $\kappa=(\kappa_1,\ldots,\kappa_n)\in\Gamma_k4$, $2k>n4$, $\kappa_14$ is the maximum entry of $\kappa4$, and for positive constants $N_04$ and $N4$ one has

$$N_0\leqslant \sigma_k(\kappa)\leqslant N.$$

For each index $1\leqslant i\leqslant n4$, if $\kappa_i>\kappa_1-\sqrt{\kappa_1}/n4$, define

$$a_j=\sigma_k^{jj}(\kappa)+(\kappa_i+\kappa_j)\sigma_k^{ii,jj}(\kappa).$$

The quadratic-form conjecture. For every $n4$-dimensional vector $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n4$, the quadratic form

$$\kappa_i\left[K\left(\sum_j\sigma_k^{jj}(\kappa)\xi_j\right)^2-\sigma_k^{pp,qq}(\kappa)\xi_p\xi_q\right]-\sigma_k^{ii}(\kappa)\xi_i^2+\sum_{j\ne i}a_j\xi_j^2$$

is nonnegative when $\kappa_14$ and the constant $K4$ are sufficiently large. Here $\sigma_k^{pp,qq}(\kappa)\xi_p\xi_q4$ denotes the repeated-index contraction of the second derivatives of $\sigma_k4$.

This conjecture is proposed as the key algebraic estimate needed for the global curvature estimate in the Hessian equation. The supplied passage gives no resolution or partial proof of the conjecture, so its status remains open.