The semilinear reducibility conjecture for zigzag graph completions

Let $ZZ_h$ be the zigzag graph with $h=h_G\geq 3$ loops, and let $\widehat{ZZ}_h$ be its completion. The notation $c_2(\widehat{ZZ}_h)_q$ denotes the $c_2$ invariant over a finite field of cardinality $q$.

Semilinear reducibility conjecture. For every $h\geq 3$, the graph $\widehat{ZZ}_h$ is semilinearly reducible and, for every $q$ with no primes of bad reduction,

$$c_2(\widehat{ZZ}_h)_q\equiv -h_G(h_G+2)\pmod q.$$

The conjecture concerns the same zigzag-completion assertion as the formula stated elsewhere in the paper: the formula is verified in the paper for $h\leq 8$, while the all-$h$ claim remains open.