The generating-function equation for graded dimensions of the free Jordan superalgebra

Let $R={\mathbb Z}[x]/(x^2-1)$, let $a_n^s=\overline{\dim}J_n(D_1|D_2)$, and write

$$a_n^s=(a_n^{\mathrm{even}},a_n^{\mathrm{odd}}).$$

Define

$$\Psi^s=\prod_{n\geq 1}\left(1-[2]_tz^n+z^{2n},0\right)^{a_n^{\mathrm{even}}}\left(\sum_{i=0}^\infty[2i+1]_tz^{2in},-\sum_{i=0}^\infty[2i+2]_tz^{(2i+1)n}\right)^{a_n^{\mathrm{odd}}},$$

and

$$\psi^s=(D_1z,D_2z)t^{-1}+(1-D_1z,-D_2z)+(-1,0)t\in R[t^{\pm1}][z].$$

The generating-function conjecture. The sequence $a_n^s$ is the unique solution of

$$\operatorname{Res}_{t=0}\psi^s\Psi^s\,dt=0.$$

This is a more explicit equation for the graded dimensions of the free Jordan superalgebra, derived in the paper from the preceding structure identities; the supplied text gives no resolution.