A proof of Gautschi's conjecture on subrange Jacobi polynomials

Let $π_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq1$, with respect to the Jacobi weight $(1-x)^α(1+x)^β$, where $-1<α<β$. Gautschi conjectured that $$ \left[ \frac{π_n(-c)}{π_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{β-α} <1. $$ By his variation formula, this inequa