Arithmetic removal lemma for cyclic groups

Question context

The arithmetic removal lemma is the following. $\forall\varepsilon>0, \exists\delta>0$ such that if $G$ is an abelian group and $A,B,C\subset G$, and suppose there are at least $\delta|G|^2$ solutions to $a+b+c=0, a\in A, b\in B, c\in C$. Then you can remove $\varepsilon|G|$ elements from each of the sets $A,B,C$ so that there are no solutions to $a+b+c=0$. In the case of $G = \mathbb{F}_p^n$ we know $\delta = \varepsilon^{c_p+o(1)}$ where $c_p$ is a known absolute constant and this is sharp. In the case of cyclic groups, we know that because of Behrend's construction $\delta$ can not be a polynomial in $\varepsilon$. The known bound is however is of tower type.

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Status note in the source

Suppose we have three sets $A=\{a_i: 1\leq i\leq n\},B=\{b_i: 1\leq i\leq n\},C= \{c_i: 1\leq i\leq n\}\subset G$. This triple is called tri-colored sumfree if $a_i+b_j+c_k=0 \iff i=j=k.$ Obtaining strong upper bounds on the size of tri-colored sum-free sets when $G$ is a cyclic group would possibly result in an improved arithmetic removal lemma.