Splitting conjecture for the filtered Yang–Baxter subcomplexes

Let $C_\bullet^{mf}$ be the relevant Yang–Baxter chain complex, and let $C_\bullet^{mf,\ell}$ be its subchain complex consisting of words using the top letter at most $\ell$ times. Consider the short exact sequence

$$0\to C_\bullet^{mf,\ell}\to C_\bullet^{mf}\to C_\bullet^{mf}/C_\bullet^{mf,\ell}\to0.$$

Filtered-subcomplex splitting conjecture. Over $k=\mathbb{Z}[y^2]$,

$$H_n(C_\bullet^{mf})=H_n(C_\bullet^{mf,\ell})\oplus H_n(C_\bullet^{mf}/C_\bullet^{mf,\ell}).$$

The claim is based on computational data and is presented as a conjectured splitting of the short exact sequence and its induced homology; no resolution is given.