The homology formula conjecture for the level-two HOMFLYPT Yang–Baxter operator

Let $R_{(2)}$ be the level-two HOMFLYPT Yang–Baxter operator, let $k$ be the coefficient ring used for the homology, and let $(f_i)$ be the delayed Fibonacci sequence. Define

$$s_{n-2}=\sum_{i=1}^{n-1}f_i=f_{n+1}-1,$$

and let $a_n$ satisfy

$$2^n=2+a_{n-1}+s_{n-3}+a_n+s_{n-2}.$$

Level-two homology formula conjecture.

$$H_n(R_{(2)})=k^2\oplus\left(k/(1-y^2)\right)^{a_n}\oplus\left(k/(1-y^2)\right)^{s_{n-2}}.$$

The supplied excerpt gives no resolution or status evidence for this formula. Its relationship to the later rank results should be checked.