Bapat's concavity conjecture for quotients of permanents

Let $b_0,b_1,\ldots,b_k$ be fixed vectors in $\mathbb{R}_{++}^n=(0,\infty)^n$, where $0\leq k<n$. For vectors $v_1,\ldots,v_n$, write $\operatorname{per}(v_1,\ldots,v_n)$ for the permanent of the matrix whose rows are these vectors. Bapat's conjecture. The function

$$x\mapsto \frac{\operatorname{per}(b_1,\ldots,b_k,x,\ldots,x)}{\operatorname{per}(b_0,b_1,\ldots,b_k,x,\ldots,x)}$$

is concave on $\mathbb{R}_{++}^n$.

The quotients generalize symmetric-function means, and the conjecture is an instance of concavity phenomena for permanents. It was proved in the source using concavity properties of hyperbolic polynomials, so the conjecture is solved.