Normalized Jones R-matrix Yang–Baxter homology conjecture

Let $V$ be the rank $2$ free $k$-module with basis $e_1,e_2$, and let $R$ be the normalized Jones R-matrix

$$R=\left(\begin{array}{cccc}1&0&0&0\\0&1-y^2&1&0\\0&y^2&0&0\\0&0&0&1\end{array}\right).$$

Write $X=(V,R)$, and let $H_n(X)$ denote its Yang–Baxter homology. 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 $a_n$ by $a_1=0$ and

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

Yang–Baxter homology conjecture. The homology groups satisfy

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

This predicts the Yang–Baxter homology of the normalized R-matrix associated with the Jones polynomial in terms of Fibonacci partial sums and the recursively specified integers $a_n$. The parser reports that the related normalization claim is proved in the cited work, but the supplied text does not establish that this homology formula itself has been resolved.