Fifth Yang–Baxter homology conjecture for the HOMFLYPT operator

Let $R_{(m)}$ be the level-$m$ Yang–Baxter operator yielding the HOMFLYPT polynomial, and let $k=\mathbb{Z}[y^2]$. The notation $\binom{m-1}{j}$ denotes a binomial coefficient. Fifth-homology conjecture.

$$\begin{aligned} H_5(R_{(m)})&\cong k^{\oplus\sum_{j=0}^{5}\binom{m-1}{j}}\\ &\oplus\left(\frac{k}{1-y^2}\right)^{\oplus\left(13\binom{m-1}{1}+124\binom{m-1}{2}+323\binom{m-1}{3}+332\binom{m-1}{4}+119\binom{m-1}{5}\right)}\\ &\oplus\left(\frac{k}{1-y^4}\right)^{\oplus\left(7\binom{m-1}{1}+16\binom{m-1}{2}+12\binom{m-1}{3}+4\binom{m-1}{4}\right)}. \end{aligned}$$

The formula is extrapolated from explicit computations of lower-level chain complexes and is presented as a computational conjecture; no proof or resolution is supplied in the source.