Monomial to ultraspherical basis transformation and the zeros of polynomials

We examine a result of A. Iserles and E. B. Saff, use it to prove a conjecture of S. Fisk that a linear operator which maps monomials to Legendre polynomials also preserves zeros in the open interval $|x|<1$, and state a more general version of the conjecture for the Jacobi polynomials.