The graded-dimension conjecture for free Jordan superalgebra components

Let $R={\mathbb Z}[x]/(x^2-1)$, and let $a_n^s=\overline{\dim}J_n(D_1|D_2)$ and $b_n^s=\overline{\dim}{\mathcal B}^s_n(J(D_1|D_2))$ be sequences in $R$. Write

$$a^s(z)=\sum_{n\geq 1}a_n^s z^n,\qquad b^s(z)=\sum_{n\geq 1}b_n^s z^n.$$

Define $\Phi^s\in R[t+t^{-1}][[z]]$ by

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

The graded-dimension conjecture. The series $a^s(z)$ and $b^s(z)$ are uniquely determined by

$$\operatorname{Res}_{t=0}(t^{-1}-1)\Phi^s\,dt=(1,0),$$ $$\operatorname{Res}_{t=0}(1-t)\Phi^s\,dt=-(D_1,D_2)z.$$

This gives generating-function identities for the graded dimensions of the homogeneous components and the associated graded object; the supplied text gives no resolution.