Corrected HZ factorisability conjecture for knots of braid index at least four

Let $\mathcal{K}$ be a knot with braid index at least $m$, where $m\geq4$. Let $H(\mathcal{K})$ be its HOMFLY–PT polynomial, let $\widehat{KF}(\mathcal{K})$ be the transformed Kauffman polynomial, and let $Z(\mathcal{K})$ be the HZ transform of $H(\mathcal{K})$. The transform is factorisable when

$$Z(\mathcal{K})=\frac{\lambda\prod_{i=0}^{m-2}(1-\lambda q^{\alpha_i})}{\prod_{i=0}^{m}(1-\lambda q^{\beta_i})}.$$

Corrected HZ factorisability conjecture. If the HOMFLY–PT/Kauffman relation holds, then the HZ transform is factorisable:

$$H(\mathcal{K})=\widehat{KF}(\mathcal{K})\implies Z(\mathcal{K})=\frac{\lambda\prod_{i=0}^{m-2}(1-\lambda q^{\alpha_i})}{\prod_{i=0}^{m}(1-\lambda q^{\beta_i})}.$$

This is proposed after the converse implication was disproved by 4-strand counterexamples; the paper expects the corrected one-way statement to hold for general $m\geq4$.