Hecke R-matrix bidifferential-complex conjecture

Let $V$ be a vector space equipped with a Hecke $\check R$-matrix, and let $K^{\bullet\bullet}(V)$ and the complexes $C_V^\bullet(k)$ be the associated bidifferential and one-variable complexes. The conditions in Theorem~ are: (i) $K^{\bullet\bullet}(V)$ decomposes as $I_{st}\oplus M$ for some $s,t\in\mathbb Z_{\geq0}$, with $M$ acyclic and $I_{st}^{n,m}=\mathbb C$ only for $(n,m)=(s,t)$; (ii) all $C_V^\bullet(k)$ with $q^{2k}=\nu_V^2$ are acyclic except one, which decomposes as $I_s\oplus N$ for some $s\in\mathbb Z_{\geq0}$, with $N$ acyclic and $I_s^n=\mathbb C$ only for $n=s$; and (iii) the natural embedding and projection between the displayed one-dimensional kernel/cohomology quotients are isomorphisms. Hecke R-matrix conjecture. Every Hecke $\check R$-matrix satisfies conditions (i)--(iii). This is presented as an open conjecture; the proof of the reverse implication in the preceding theorem is explicitly omitted and attributed to a classification result not established in the paper.