The structure conjecture for the free Jordan superalgebra $J(D_1|D_2)$

Let $J(D_1|D_2)$ be the free Jordan superalgebra with $D_1+D_2\geq 1$. Let $G$ be the acting algebraic group, let ${\mathcal R}^s(G)$ be the relevant super representation ring, let $\lambda^s$ be the super $\lambda$-operation, and let $L(0)$ and $L(2)$ denote the indicated $PSL(2)$-modules. Set

$$A(D_1|D_2)=[J(D_1|D_2)],\qquad B(D_1|D_2)=[{\mathcal B}^s(J(D_1|D_2))].$$

The structure conjecture. The elements $A(D_1|D_2)$ and $B(D_1|D_2)$ are uniquely determined in ${\mathcal R}^s(G)$ by

$$[\lambda^s(A(D_1|D_2)L(2)+B(D_1|D_2)):L(0)]=[k],$$ $$[\lambda^s(A(D_1|D_2)L(2)+B(D_1|D_2)):L(2)]=-[k^{D_1|D_2}],$$

where $k$ is the trivial graded $G$-module and $k^{D_1|D_2}$ is the natural graded $G$-module. This is the proposed representation-ring description of the free Jordan superalgebra and its associated object; the supplied text gives no resolution.