The arbitrary-group induced removal conjecture

Fix a bipartite graph $F$, and let $G$ be a finite group. A subset $A \subseteq G$ is $\epsilon$-far from bi-induced-$F$-free if changing fewer than $\epsilon |G|$ elements of $A$ cannot make it bi-induced-$F$-free. A uniform random map is a map $\phi \colon V(F) \to G$ chosen uniformly from all such maps, and it bi-induces $F$ when

$$\phi(u)+\phi(v) \in A \quad\Longleftrightarrow\quad uv \in E(F)$$

for every edge-pair $(u,v) \in U \times V$ of the bipartition of $F$.

Arbitrary-group induced removal conjecture. For every $0 < \epsilon < 1/2$, if $A \subseteq G$ is $\epsilon$-far from bi-induced-$F$-free, then

$$\Pr_{\phi \colon V(F) \to G}[\phi\text{ bi-induces }F] \geq \epsilon^{O_F(1)}.$$

This would extend the removal lemma from abelian groups to arbitrary groups. The claim is stated as open even for general abelian groups, and therefore remains open in the full arbitrary-group setting.