The weak form of Conjecture 1 for free Jordan algebras

Let $D\geq 1$, let $J(D)=\bigoplus_{n\geq 1}J_n(D)$ be the free Jordan algebra, and let $\operatorname{Inner}_nJ(D)$ denote the degree-$n$ component of its inner derivation algebra. Let $a_n(D)$ and $b_n(D)$ be the uniquely defined coefficients from Lemma.

Weak form of Conjecture 1.

$$\dim J_n(D)=a_n(D),\qquad \dim\operatorname{Inner}_nJ(D)=b_n(D).$$

This is the dimension-level consequence of the proposed character formula. The paper gives computational evidence for the conjecture, but no general proof.