Currying of an irreducible positive-degree plane polynomial is primitive

Let $h$ be an irreducible bivariate real polynomial ($\mathrm{Irreducible}\,h$) whose currying has positive degree, $0 < (\mathrm{Curry}_0(h)).\mathrm{natDegree}$. Then the curried univariate polynomial over the coefficient ring is primitive:

Primitivity (content a unit) is the hypothesis needed for Gauss-type arguments: it allows coprimality over the coefficient ring to be transported to the fraction field, where the resultant nonvanishing argument runs.