The weakest dimension conjecture for free Jordan algebras

Let $K$ be a field of characteristic zero, let $J(D)$ be the free Jordan algebra on $D$ generators, and write $J(D)=\bigoplus_{n\geq 1}J_n(D)$. Set $a_n=\dim J_n(D)$ and let $\psi=Dzt^{-1}+(1-Dz)-t$.

The weakest dimension conjecture. The sequence $(a_n)_{n\geq 1}$ is the unique solution of

$$\operatorname{Res}_{t=0}\,\psi\prod_{n=1}^{\infty}\left(1-z^n(t+t^{-1})+z^{2n}\right)^{a_n}\,dt=0.$$

The equation gives a recurrence determining the predicted dimensions, and computer calculations support it in several cases, including the dimension three space of special identities for $D=3$ and $n=8$. A closed formula is not known.