Polynomial-dimension conjecture for high additive energy in $F_3^N$

Let $N>0$ be a large parameter, with constants independent of $N$. Let $B,C\subseteq F_3^N$ satisfy $|B|,|C|\leq N^{100}$. Suppose that the number of quadruples $(b_1,c_1,b_2,c_2)\in B\times C\times B\times C$ satisfying

$$b_1+c_1=b_2+c_2$$

is at least $|B||C|^2N^{-\sigma}$. High-energy subspace conjecture. Then $C$ is contained in a subspace of dimension $N^{O(\sigma)}$. This is the first of two conjectures proposed as possible improvements to the efficiency of the paper's iterative argument; the source does not state a resolution.