Level-six torsion conjecture for the HOMFLYPT Yang–Baxter operator

Let $R_{(m)}$ be the level-$m$ Yang–Baxter operator yielding the HOMFLYPT polynomial, let $k=\mathbb{Z}[y^2]$, and let $a_6(m),b_6(m),c_6(m),d_6(m)$ denote the multiplicities in the proposed decomposition. Level-six torsion conjecture. Over $k=\mathbb{Z}$ with $y=2$,

$$H_6(R_{(m)},\mathbb{Z})=\mathbb{Z}^{a_6(m)}\oplus\mathbb{Z}_3^{b_6(m)}\oplus\mathbb{Z}_{15}^{c_6(m)}\oplus\mathbb{Z}_{315}^{d_6(m)}.$$

Over $k=\mathbb{Z}[y^2]$,

$$H_6(R_{(m)})=k^{a_6(m)}\oplus\left(\frac{k}{1-y^2}\right)^{b_6(m)}\oplus\left(\frac{k}{1-y^4}\right)^{c_6(m)}\oplus\left(\frac{k}{(1-y^2)(1+y^2)(1+y^2+y^4)}\right)^{d_6(m)}.$$

The conjecture is motivated by computations, including the observed additional torsion factor; the source only notes the lower bound $d_6(m)\geq m(m-1)$ and does not resolve the full decomposition.