realized.

Although being technically more involved, the second part of the proof is fairly similar to the corresponding arguments in [@MM5; @MM6; @Zi; @MaMa]. For Theorem \[thmintro\], an additional essential part is the construction of the two extra simple transitive $2$-representations. This construction follows closely the approach of [@MaMa Section 5]. Apart from this, the main general difficulty in the proof of Theorem \[thmintro\] lies in the first part of the proof. Both the difficulty of this part and the approach we choose to deal with this part depends heavily on each particular Coxeter type we study.

For Coxeter groups of rank higher than two, we use a reduction argument to the rank two case. For this reduction to work, we need the statement of Theorem \[thmintro\] in Coxeter types $I_2(n)$, where $n=3,4,5$. For $n=3,5$, we may directly refer to Theorem \[thmintro\]. As already mentioned above, the case $n=4$ is treated in [@Zi]. Behind this reduction is the observation that Soergel bimodules corresponding to the longest elements in rank two parabolic subgroups of $W$ do not survive projection onto $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$.

The argument for the first part of the proof which we employ in the rank two case is based on 