The concavifying-function conjecture for Lovelock symmetric functions

Let $\overrightarrow{a}=(a_0,a_1,\ldots,a_p)\in(\mathbb{R}_+)^{p+1}$ and define

$$f_{\overrightarrow{a}}:=a_0\sigma_0+a_1\sigma_1+\cdots+a_p\sigma_p.$$

Here $\Gamma_n$ denotes the Gårding cone. Concavifying-function conjecture. There exists $F\in\mathcal{C}^2(\mathbb{R}_+^*\to\mathbb{R})$ such that $F\circ f_{\overrightarrow{a}}$ is concave on $\Gamma_n$. The conjecture would extend the paper's existence of concavifying functions beyond the cases established there, while the general assertion remains unproved.