Conjectural irreducibility criterion for the Lawrence–Krammer representation

Let $G_1,\ldots,G_{n-1}$ be the matrices describing the actions of the BMW generators $g_1,\ldots,g_{n-1}$ on the spanning elements of a vector space $W$, and define

$$E_i=\frac{l}{m}\left(G_i^2+mG_i-I_{\binom{n}{2}}\right),\qquad 1\leq i\leq n-1.$$

Here $B(A_{n-1})$ is the BMW algebra, $F$ is the coefficient field, and $\mathcal{M}(k,F)$ denotes the $k\times k$ matrices over $F$. Irreducibility conjecture. For $n=3$, the assignment

$$B(A_2)\longrightarrow\mathcal{M}(3,F),\qquad g_1,g_2\longmapsto G_1,G_2,\qquad e_1,e_2\longmapsto E_1,E_2$$

is a representation, and it is irreducible if and only if $l\notin\\{-r^3,-1,1,\frac{1}{r^3}\\}$. In general, the assignment

$$B(A_{n-1})\longrightarrow\mathcal{M}\left(\binom{n}{2},F\right),\qquad g_i\longmapsto G_i,\qquad e_i\longmapsto E_i$$

is a representation, and it is irreducible if and only if $l\notin\\{r,-r^3,-\frac{1}{r^{n-3}},\frac{1}{r^{n-3}},\frac{1}{r^{2n-3}}\\}$. No resolution is supplied in the cited passage.