Ruzsa's genus conjecture

Consider a linear equation

$$E:\sum_{i=1}^k a_i x_i=0,$$

where $a_1,\dots,a_k$ are integers. The equation has genus one if

$$\sum_{i=1}^k a_i=0$$

and $\sum_{i\in T}a_i\neq 0$ for every nonempty proper subset $T\subsetneq [k]$. Let $r_E(m)$ be the maximum size of a subset of $[m]$ containing no non-trivial solution to $E$, where a non-trivial solution is a solution whose variables are not all equal. Ruzsa's genus conjecture. If $E$ is a linear equation of genus one, then

$$r_E(m)\geq m^{1-o(1)}.$$

This is a conjecture in additive combinatorics about the size of subsets avoiding non-trivial solutions to genus-one equations; the supplied source does not state whether it has been resolved.