to all cases we consider and substantially simplifies some of the arguments. For example, in Coxeter type $I_2(5)$, we originally had an independent argument for a similar result which involved a very technical statement that a certain collection of twenty seven linear inequalities is equivalent to the fact that all these inequalities are, in fact, equalities. This full argument was three pages long and just covered one Coxeter type. A similar argument in other Coxeter types of the form $I_2(n)$ seemed, for a long period of time, unrealistic.

Our results form a first step towards classification of simple transitive $2$-representations of Soergel bimodules in all types. However, for the moment, the technical difficulty of solving the first part of the problem (as described above) in the general case seems too high. Already in type $B_3$ the classification of simple transitive $2$-representations is not complete. It is also of course very natural to ask what happens in positive characteristics. However, the situation there is expected to be even more complicated.

The paper is organized as follows: In Section \[s2\] we collect all necessary preliminaries on $2$-categories and $2$-representations. Section \[s3\] collects preliminaries on $2$-categories of Soergel bimodules. In Section \[s4\] we 