Dual Ore conjecture for distributive intervals of finite groups

Let $[H,G]$ be an interval of finite groups. For a complex representation $W$ of $G$ and a subgroup $K$, write

$$W^{K}:=\{w\in W\mid kw=w\text{ for all }k\in K\}$$

and, for a subspace $X$ of $W$, write

$$G_{(X)}:=\{g\in G\mid gx=x\text{ for all }x\in X\}.$$

An interval $[H,G]$ is linearly primitive if there is an irreducible complex representation $V$ of $G$ such that $G_{(V^H)}=H$.

Dual Ore conjecture. Every distributive interval $[H,G]$ of finite groups is linearly primitive; equivalently, there exists an irreducible complex representation $V$ of $G$ such that

$$G_{(V^H)}=H.$$

This is the proposed dual version of Ore's theorem, which characterizes finite groups with distributive subgroup lattices as cyclic. The paper proves the assertion for distributive intervals of index less than $9720$ and for boolean intervals of rank less than $7$; the general statement remains open in the supplied source.