The dimension-generating-series conjecture for free alternative algebras

Let $D$ be a non-negative integer, let $A(D)$ be the free alternative algebra with $D$ generators, and let $\mathcal B(A(D))$ denote its inner algebra. Define

$$a(z)=\sum_{n\geq1}a_nz^n,\qquad b(z)=\sum_{n\geq1}b_nz^n\in z\mathbb Z[[z]].$$

Let $\Delta$ be the root system used in the source, and define

$$\Phi=\prod_{n\geq1}\left(\prod_{\alpha\in\Delta}(1-e^\alpha z^n)^{a_n}\right)(1-z^n)^{2a_n+b_n}$$

in $\mathbb Z[[z]][t_1^{\pm1},t_2^{\pm1}]^{S_3}$.

Dimension-generating-series conjecture. Suppose $a(z)$ and $b(z)$ are the solutions of

$$\Phi(0)=1,\qquad \Phi(\alpha_1+\alpha_2)=-Dz.$$

Then $a(z)$ and $b(z)$ are generated from the dimensions of the degree-$n$ homogeneous components of $A(D)$ and $\mathcal B(A(D))$.

This is the numerical generating-series form of the Grothendieck-class conjecture. The paper derives the equations from the proposed representation-theoretic description, but does not prove that the resulting coefficients give the dimensions in all degrees.