Ren–Wang's concavity conjecture for elementary symmetric functions

Let $\kappa=(\kappa_1,\ldots,\kappa_n)\in\Gamma_k$ satisfy $\kappa_1\geq\kappa_2\geq\cdots\geq\kappa_n$ and $n<2k$. Assume that there are constants $N_0,N_1$ such that

$$N_0\leq\sigma_k(\kappa)\leq N_1.$$

If there are constants $K$ and $B$ such that $\kappa_1\geq B$, define

$$a_j=\sigma_k^{jj}(\kappa)+(\kappa_1+\kappa_j)\sigma_k^{11,jj}(\kappa).$$

Ren–Wang's concavity conjecture. For every $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n$,

$$\kappa_1\left(K\left(\sum_j\sigma_k^{jj}(\kappa)\xi_j\right)^2-\sigma_k^{pp,qq}(\kappa)\xi_p\xi_q\right)-\sigma_k^{11}(\kappa)\xi_1^2+\sum_{j\neq 1}a_j\xi_j^2\geq 0.$$

This conjectured inequality is intended to provide the key algebraic estimate needed for curvature estimates for prescribed $k$-th curvature equations when $n<2k$. It was previously used in work of Ren and Wang and is being proposed here as a step toward extending their Euclidean curvature estimates to warped product manifolds; its resolution is not supplied in the source.