Spin-parity refinement of the polynomiality conjecture for Witten's classes

Let $a$ be fixed, and suppose that $r$ is even and $a$ is odd. Let $W_g^\pm(a,r)$ denote the refinement of Witten's class according to spin parity, and let $[\overline{\mathcal{M}}_g(a)]^\pm$ denote the corresponding spin-parity refinement of the class of the stratum of holomorphic differentials. Spin-parity polynomiality conjecture. For fixed $a$, the class $W_g^\pm(a,r)$ behaves polynomially in $r$, and $[\overline{\mathcal{M}}_g(a)]^\pm$ is the constant coefficient of this polynomial. This conjecture is the spin-parity refinement of the polynomiality and constant-term relation for Witten's classes, providing a proposed method for computing classes of strata of holomorphic differentials with prescribed spin parity. Its status is not determined in the supplied source.