The real-rootedness criterion for concavity of symmetric-polynomial sums

Let $n,p\in\mathbb{N}^*$, let $\Gamma_n$ be the positive cone, and let $\overrightarrow{a}=(a_0,\ldots,a_p)\in(\mathbb{R}_+)^{p+1}$. Set

$$f_{\overrightarrow{a}}=a_0+a_1\sigma_1+\cdots+a_p\sigma_p,$$

and let its diagonal restriction be

$$\bar f_{\overrightarrow{a}}(X)=\sum_{k=0}^p a_k\binom{n}{k}X^k.$$

Real-rootedness concavity conjecture. If $\bar f_{\overrightarrow{a}}$ is real-rooted, then $f_{\overrightarrow{a}}$ is $\frac1p$-concave on $\Gamma_n$; equivalently, $\Xi_n^p\subset\mathcal{K}_n^p$. This is presented as a weaker version of the diagonal concavity conjecture and remains unproved in the paper.