The diagonal concavity conjecture for sums of elementary symmetric polynomials

Let $n,p\in\mathbb{N}^*$, let $\Gamma_n$ be the positive cone, let $\Delta=\{(X,\ldots,X):X>0\}$ be its positive diagonal, and let

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

where $\overrightarrow{a}=(a_0,\ldots,a_p)\in(\mathbb{R}_+)^{p+1}$. Define the diagonal restriction

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

Diagonal concavity conjecture. The function $f_{\overrightarrow{a}}$ is $\frac1p$-concave on $\Gamma_n$ if and only if it is $\frac1p$-concave on $\Delta$, if and only if $\bar f_{\overrightarrow{a}}$ is $\frac1p$-concave on $\mathbb{R}_+^*$. Equivalently, $\mathcal{K}_n^p=\mathcal{X}_n^p$. The conjecture is explicitly stated as unproved in the paper, although equivalence is established in some cases.