Conjectural composition factors of the third lower-central quotient of the symplectic free associative algebra

Let $A'_{2n}$ be the symplectic quotient of the free associative algebra on $2n$ generators, let $B_3(A'_{2n})$ denote its third lower-central-series quotient, and let $H_{2n}$ act on it. Write $\rho_m$ for the relevant irreducible $\tfrac{\mathfrak{sp}}{}_{2n}$-representation, and let $\mathcal{F}_\lambda$, $T_k$, $X_k$, and $Z_k$ denote the tensor-field modules and subquotients defined in the paper.

Composition-factor conjecture. The vectors $v_{k,0}$ and $z_{k,0}$ are non-zero in $B_3(A'_{2n})$, and the $H_{2n}$-module composition factors of $B_3(A'_{2n})$ are

$$\mathcal{F}_{(2,1^k)},\,\mathcal{F}_{(1^k)}/T_k,\,Z_k/X_k$$

for odd $k$ with $1\le k\le n-1$.

This conjecture is based on direct computations in MAGMA for $2n=4,6$ and would identify the displayed surjection onto $B_3(A'_{2n})$ as an isomorphism for all $n$.