transitive $2$-representations for the $2$-category of Soergel bimodules in Weyl type $B_2$. In [@MZ], a similar result was proved for the $2$-categories of projective functors for the two smallest non-self-injective finite dimensional algebras. Some other related results can be found in [@GM1; @GM2; @Zh1; @Zh2].

An essential novel step in this theory was made in the recent paper [@MaMa] where a classification of the simple transitive $2$-representations was given for certain $2$-subquotient categories of Soergel bimodules in two dihedral Coxeter types. In one of the cases, it turned out that cell $2$-representations do not exhaust the simple transitive $2$-representations. A major part of [@MaMa] is devoted to an explicit construction of the remaining simple transitive $2$-representation. This construction involves a subtle interplay of various category theoretic tricks.

The present paper explores to which extent the techniques developed in [@MM5; @Zi; @MaMa] can be used to attack the problem of classification of the simple transitive $2$-representations for $2$-categories of Soergel bimodules over coinvariant algebras in the general case of finite Coxeter systems. We develop the approach and intuition described in [@Zi; @MaMa] further and single out a situation in which this approach seems to be applicable. The combinatorial structure of the 