Odd-even polynomial-integral conjecture for natural systems on the torus

Let $T^2$ be the two-dimensional torus, and let a natural Hamiltonian system on it have a nonconstant potential. A linear, quadratic, or higher-degree polynomial integral is an integral polynomial in the momenta of the indicated degree. Odd-even integral conjecture. If the system admits an integral polynomial of odd degree in momenta, then it admits a linear integral; if it admits a nontrivial integral of even degree, then it admits a nontrivial quadratic integral. This is described as an easier version of the preceding torus conjecture, and the source does not state that it has been resolved.