The third Yang–Baxter homology formula for the Homflypt rack family

Let $R_{(m)}$ be the rack considered in the paper, let $H_3(R_{(m)})$ denote its third Yang–Baxter homology over $k$, and let $m$ be the parameter indexing this rack family. The third Yang–Baxter homology conjecture. The homology group is conjectured to satisfy

$$H_3(R_{(m)})=k^{\frac{m(8-3m+m^2)}{6}}\bigoplus \left(k/(1-y^2)\right)^{\frac{(m^2-1)(5m-6)}{6}}\bigoplus \left(k/(1-y^4)\right)^{m(m-1)}.$$

This formula is suggested by the computed values in the $H_3$ row of the table for $m=3,4,5,6,7$. No general proof or resolution is given in the source.