Let zn,j(λ) denote the positive zeros, in decreasing order, of the ultraspherical polynomial Cλn, λ>−1/2, with the reduced limiting interpretation at λ=0 specified below. Our principal result settles three higher-monotonicity questions of Gautschi: two as printed and the natural open-interval form of the third, whose printed endpoint λ=0 is singular. For every n≥3,

becomes a complete Bernstein function after translation of its parameter interval to (0,∞). For every n≥2,

is a complete Bernstein function on (0,∞). For every n≥4, the largest-zero trajectory

has the same property, whilst for every other positive zero the derivative of this scaling fails to be completely monotone. Separately, an exact calculation in degree 4 provides a counterexample to a fourth conjecture of Gautschi, concerning the linear scaling λzn,j(λ).