Yang–Baxter homology conjecture for the level-two operator

Let $R_{(2)}$ be the level-two Yang–Baxter operator, let $H_n(R_{(2)})$ denote its $n$th homology, and let $k$ be the coefficient ring. Let $f_n$ be the Fibonacci numbers, with $f_0=0$ and $f_1=1$. Level-two homology conjecture.

$$H_n(R_{(2)})=k^2\oplus\left(\frac{k}{1-y^2}\right)^{\oplus b_n(2)}\oplus\left(\frac{k}{1-y^4}\right)^{\oplus c_n(2)},$$

where

$$c_n(2)=f_{n+1}-1,\qquad b_n(2)=\frac{2^{n+1}+(-1)^n}{3}-f_{n+1}.$$

The claim is part of the paper’s program of predicting further Yang–Baxter homology calculations for the operators yielding the HOMFLYPT polynomial; the source provides no resolution.