Künneth subcomplex splitting conjecture for Yang–Baxter chain complexes

Fix a decomposition of the alphabet $X_{(m)}=A\cup B$ and let $C_\bullet^{(m,A,B)}$ be the subchain complex generated by words that begin with letters from $A$ and end with letters from $B$. Assume the ordering condition $A\geq B$. Künneth splitting conjecture. The short exact sequence

$$0\to C_\bullet^{(m,A,B)}\to C_\bullet^m\to C_n^m/C_\bullet^{(m,A,B)}\to0$$

splits, and consequently $H_n(C_\bullet^m)$ splits. This conjecture is motivated by the identification of the subcomplex with a tensor product of smaller chain complexes and by computational evidence; the source gives no proof or resolution.