Sturmfels–Zworski conjecture on cyclic resultants of reciprocal polynomials

Let $K$ be the coefficient field, and let $f(x)\in K[x]$ be a reciprocal monic polynomial of even degree $d$, meaning that $f(1/x)=x^d f(x)$. Its cyclic resultants are the associated sequence of resultants. Sturmfels–Zworski conjecture. A reciprocal monic polynomial $f(x)\in K[x]$ of even degree $d$ is determined by its first $d/2+1$ cyclic resultants. The conjecture concerns the amount of period or resultant data needed to reconstruct reciprocal polynomials. The paper notes progress for a special class arising from zeta functions of curves over finite fields, but the stated conjecture remains open in general.