Polynomials nonnegative on the cylinder

In 2010, Marshall settled the strip conjecture, according to which every polynomial in $\mathbb{R}[x,y]$, nonnegative on the strip $[-1,1]\times\mathbb{R}$, is a sum of squares and of squares times $1-x^2$. We consider affine nonsingular curves $C$ over $\mathbb{R}$ with $C(\mathbb{R})$ compact, and study the question whether every $f$ in $\mathbb{R}[C][y]$, nonnegative on $C(\mathbb{R})\times\mathbb{R}$, is a sum of squares in $\mathbb{R}[C][y]$. We give an affirmative answer under the condition that $f$ has only finitely many zeros in $C(\mathbb{R})\times\mathbb{R}$. For $C$ the circle $x_1^2+x_2^2=1$, we prove the result unconditionally.