$\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ has two extra equivalence classes of simple transitive $2$-representations, see the explicit construction in Subsection \[s6.7\]. If $n\neq 12,18,30$, these are all the simple transitive $2$-representations.

We note that Coxeter type $I_2(4)$ is dealt with in [@Zi] and the result in this case is similar to Theorem \[thmintro\]. In fact, the construction in Subsection \[s6.7\] also works in type $I_2(4)$, however, it results in $2$-representations which turn out to be equivalent to cell $2$-representations. The exceptional types $I_2(12)$, $I_2(18)$ and $I_2(30)$ exhibit some strange connection, which we do not really understand, to Dynkin diagrams of type $E$ that, at the moment, does not allow us to complete the classification of simple transitive $2$-representations in these types.

Theorem \[thmintro\] is proved by a case-by-case analysis. Similarly to the general approach of [@MM5; @MM6; @Zi; @MaMa], in each case, the proof naturally splits into two major parts:

-   the first part of the proof determines the non-negative integral matrices which represent the action of Soergel bimodules corresponding to simple reflections;

-   to each particular case of matrices determined in the first part, the second part of the proof provides a classification of simple transitive $2$-representations for which these particular matrices are 