Schur-functor decomposition conjecture for lower central series quotients

Let $T(V)$ be the tensor algebra of a vector space $V$, and let $B_k(T(V))$ be its $k$th lower central series quotient. For a Young diagram $\lambda$, let $\mathcal{G}_\lambda$ denote the corresponding Schur functor occurring in the paper. Schur-functor decomposition conjecture. Let $k\geq 3$. There exists a finite collection $\Lambda$ of Young diagrams $\lambda$, and an isomorphism of abstract Schur functors

$$V\mapsto B_k(T(V))\cong \bigoplus_{\lambda\in\Lambda}\mathcal{G}_\lambda.$$

This would imply the rationality conjecture above. It strengthens the preceding structural bound and remains open in the general setting.