Strict convexity conjecture for the general inverse $\sigma_k$ equation

Consider the general inverse $\sigma_k$ equation

$$f(\lambda)\coloneqq\sigma_n(\lambda)-\sum_{k=0}^{n-1}c_k\sigma_k(\lambda)=0.$$

Let $r_f(x)$ be the diagonal restriction of $f(\lambda)$, and call it strictly right-Noetherian when it has the strict right-Noetherian property used in the paper. On solving the equation for $\lambda_n$, write

$$\lambda_n=\frac{\sum_{k=0}^{n-1}c_k\sigma_k(\lambda_{;n})}{\lambda_1\cdots\lambda_{n-1}-\sum_{k=0}^{n-1}c_k\sigma_{k-1}(\lambda_{;n})}.$$

Strict convexity conjecture. If $r_f(x)$ is strictly right-Noetherian, then the level set $\{f=0\}$ is strictly convex; equivalently, the Hessian matrix

$$\left(\frac{\partial^2\lambda_n}{\partial\lambda_i\partial\lambda_j}\right)_{i,j\in\{1,\ldots,n-1\}}$$

is positive-definite on the level set $\{f=0\}$. The preceding theorem proves convexity of the level set, while strict convexity remains open in the stated generality.