Le's conjecture for symmetric hyperbolic polynomials

Let $P$ be a homogeneous, real, symmetric polynomial of degree $k$ on $\mathbb{R}^n$ that is hyperbolic with respect to $\mathbf{1}$. Let $A\in\mathbb{S}_n$, write $\lambda(A)$ for the vector of eigenvalues of $A$, and let $\operatorname{diag}(A)$ denote its diagonal vector. For a hyperbolic polynomial $P$, let $\Gamma(P)$ be its hyperbolicity cone with respect to $\mathbf{1}$. Suppose that

$$\lambda(A)\in\Gamma(P).$$

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

$$P(\operatorname{diag}(A))\ge P(\lambda(A)).$$

This is a Schur--Horn type principle relating eigenvalue majorization to concavity on hyperbolicity cones. The supplied text attributes the conjecture to Le, but gives no evidence resolving it.