so-called [*finitary*]{} $2$-categories which are natural $2$-analogues of finite dimensional algebras. The penultimate paper [@MM5] of this series proposes the notion of a [*simple transitive $2$-representation*]{} which seems to be a natural $2$-analogue for the classical notion of a simple module. Furthermore, [@MM5; @MM6] classifies simple transitive $2$-representations for a certain class of finitary $2$-categories with a weak involution which enjoy particularly nice combinatorial properties, the so-called [*(weakly) fiat*]{} $2$-categories with [*strongly regular two-sided cells*]{}. Examples of the latter $2$-categories include projective functors for finite dimensional self-injective algebras, finitary quotients of finite type $2$-Kac-Moody algebras and Soergel bimodules (over the coinvariant algebra) in type $A$. The classification results of [@MM5; @MM6] assert that, for such $2$-categories, every simple transitive $2$-representation is, in fact, equivalent to a so-called [*cell $2$-representation*]{}, that is a natural subquotient of the regular (principal) $2$-representation defined combinatorially in [@MM1; @MM2]. In [@KM1], this classification was used to describe projective functors on parabolic category $\mathcal{O}$ in type $A$.

The problem of classifying simple transitive $2$-representations was recently studied for several classes of finitary $2$-categories which are not covered by the results in [@MM5; @MM6]. In particular, in [@Zi] it is shown that cell $2$-representations exhaust the simple 