TITLE: A dual version of a theorem of Øystein Ore in group theory
QUESTION [12 upvotes]: This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem:  Let $[H, G]$ be a  distributive interval of finite groups. Then $\exists g \in G$ such that $\langle H,g \rangle = G$.  
Definition: Let $W$ be a representation of $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
 Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \  kw=w \ , \forall k \in K  \}$.
Let the pointwise stabilizer subgroup $G_{(X)}:=\{ g \in G \  \vert \ gx=x \ , \forall x \in X \}$. 
Definition:  $[H,G]$ is called linearly primitive if $\exists V$ irred. complex repr. of $G$ with $G_{(V^H)} = H$. 
 Remark: $[1,G]$ is linearly primitive iff $G$ is linearly primitive.
Question: Let $[H, G]$ be a distributive interval of finite groups. Is $[H,G]$ linearly primitive?  
Remark: The case $H = 1$ is true because $[1,G]=\mathcal{L}(G)$ is distributive iff $G$ is cyclic, but an abelian group is linearly primitive iff it is cyclic. It follows that it's also true if $H \triangleleft G$.   
It's true by GAP for $|G:H|<32$ or $\vert G \vert \le 1000$ or $G$ perfect with $\vert G \vert < 10080$ (except $7680$).

REPLY [2 votes]: Yes. 
This was proved in the planar algebra framework, see arXiv:1704.00745, Corollary 6.10.    
For a self-contained group-theoretic proof, see arXiv:1708.02565.