Conjecture on finite-dimensional elementary Fomin–Kirillov Nichols algebras

Let $G=G(m,p,n)$ be a complex reflection group in the non-exceptional series, and let $\mathcal{B}_G$ be its elementary Fomin–Kirillov Nichols algebra. Finite-dimensionality conjecture. Up to isomorphism, the only finite-dimensional elementary Fomin–Kirillov Nichols algebras $\mathcal{B}_G$ are those associated with $G=S_2,S_3,S_4,S_5$. Hence, the only finite-dimensional Fomin–Kirillov Nichols algebras for complex reflection groups $G(m,p,n)$ are those of $S_2\cong C_2$, $S_3\cong \operatorname{Dih}_3$, $S_4\cong D_3$, $S_5$, $B_2\cong \operatorname{Dih}_4$, $C_3$, $C_4$, and $D_2\cong S_2\times S_2$. The conjecture is derived from the results discussed in the paper and a cited conjecture, so its general validity remains open.