Jain's incidence-code parameter conjecture for unit graphs

Let $n$ be a natural number, let $G(\mathbb{Z}_n)$ be a unit graph, and let $H$ be a $|V|\times|E|$ incidence matrix of $G(\mathbb{Z}_n)$. Let $\phi(n)$ denote Euler's totient function, and let $C_q(H)$ denote the code generated by $H$ over the finite field $\mathbb{F}_q$. Jain's incidence-code conjecture. If $2\in U(\mathbb{Z}_n)$, then the binary code generated by $H$ is

$$C_2(H)=\left[\frac{(n-1)\phi(n)}{2}, n-1, \phi(n)-1\right]_2$$

over $\mathbb{F}_2$. If $2\in N_U(\mathbb{Z}_n)$, then, for any odd prime $q$, the $q$-ary code generated by $H$ is

$$C_q(H)=\left[\frac{n\phi(n)}{2}, n-1, \phi(n)\right]_q$$

over $\mathbb{F}_q$.

This conjecture specifies the length, dimension, and minimum distance of incidence-matrix codes associated with unit graphs. The surrounding paper states that it resolves this coding-theoretic conjecture from Jain2023, so this candidate is treated as solved.