Fisk's Legendre zero-preservation conjecture

Let $f(x)=\sum_{k=0}^n a_kx^k\in\mathbb{R}[x]$ have all its zeros in $(-1,1)$, and define the linear transformation $T$ by

$$T[f](x)=\sum_{k=0}^n a_kP_k(x),$$

where $P_k(x)$ is the $k$-th Legendre polynomial. Fisk's conjecture. The polynomial $T[f](x)$ has all its zeros in $(-1,1)$. The paper states that this conjecture is proved there; it had previously been left open in Fisk's book.