Degree conjecture for surfaces of revolution with trigonometric-polynomial support functions

Let $N\geq 2$, and let $p$ be a trigonometric polynomial of degree $N$ with coefficients $a_N,b_N$ satisfying $a_N\ne 0$ or $b_N\ne 0$. Let $f\in\mathbb{R}[x,y,z]$ be the defining polynomial of the surface of revolution obtained by rotating the curve associated with $p$ around the $X$-axis. Degree conjecture. The partial degrees of $f$ satisfy

$$\deg_x f=\deg_y f=\deg_z f=4N+4,$$

and the total degree of $f$ is $4N+4$. In particular, a constant-width surface of revolution has a defining polynomial of degree $4N+4$ when its support function is a trigonometric polynomial of degree $N$. The claim is motivated by numerical computations and proposes a simple relation between the support-function degree and the defining-polynomial degree; the source does not state whether it has been proved or disproved.