Homology decomposition conjecture for the two-term Yang–Baxter complex

Let $k=\mathbb{Q}[y,y^{-1}]$, let $V=k\{v_1,v_2\}$, and let $H_n$ denote the $n$-th homology group of the two-term Yang–Baxter chain complex associated with the displayed unital Yang–Baxter operator for $m=2$. Define $s_n=\sum_{i=1}^{n+1}f_i$, where $f_1=f_2=1$ and $(f_i)$ is the Fibonacci sequence, and define $a_n$ recursively by

$$2^n=2+a_{n-1}+s_{n-2}+a_n+s_{n-1},\qquad a_1=0.$$

Homology decomposition conjecture. For every $n$,

$$H_n\cong k^2\oplus\left(k/(1-y^2)\right)^{a_n}\oplus\left(k/(1-y^4)\right)^{s_{n-2}}.$$

The formula was verified in the source for $n\leq 10$ and predicts the structure of the two-term Yang–Baxter homology in this family beyond the checked range.