The integral factorisation conjecture for the fused Hecke algebra

Let $\mathcal{A}_{k+1}$ be the algebra generated by the generators and relations of Definition of the paper, and let $E_{k+1}^{(q,\alpha_1)}$ be the corresponding element. Specialise $(\alpha_1,\alpha_2)$ to $(q^{-2},q^{2k})$, and write $[k+1]_q!$ for the quantum factorial. Integral factorisation conjecture. In $\mathcal{A}_{k+1}$, the specialised element factorises as

$$E_{k+1}^{(q,\alpha_1)}=[k+1]_q!\,\widetilde{E}_{k+1}^{(q,\alpha_1)},$$

where $\widetilde{E}_{k+1}^{(q,\alpha_1)}$ belongs to $\mathcal{A}_{k+1}$ with coefficients in $\mathbb{C}[q^{\pm1}]$. This conjectural factorisation is intended to provide the correct definition of the algebra over $\mathbb{C}[q^{\pm1}]$, where the unreduced presentation does not have the required freeness and dimension properties.