The pairing-matrix finite-order reformulation of the zero divisor conjecture

Let $C$ be a pairing matrix, meaning an integer $n\times m$ matrix $C=(c_{ij})$ such that $nm$ is even, $c_{i,j}\in\{1,2,\ldots,nm/2\}$, no row or column contains a repeated value, and every pair $(i,j)$ has exactly one distinct partner $(i',j')$ with $c_{ij}=c_{i'j'}$. Define

$$\Gamma_C=\langle a_1,\ldots,a_n,b_1,\ldots,b_m\mid a_ib_j=a_{i'}b_{j'}\text{ whenever }c_{ij}=c_{i'j'}\rangle.$$

Pairing-matrix conjecture. If the generators $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ of $\Gamma_C$ are distinct, then $\Gamma_C$ contains a nontrivial element of finite order. This is presented as an equivalent reformulation of the zero divisor conjecture over ${\mathbb F}_2$ and is stated as open in the supplied context.