Voss's conjecture on chords in odd cycles of critical graphs

All graphs are finite and simple. A graph is $k$-critical if its chromatic number is $k$ and every proper subgraph is $(k-1)$-colorable. Let $g_k(n)$ be the largest integer $\alpha$ such that every $n$-vertex $k$-critical graph, with $k\geq4$, contains an odd cycle with at least $\alpha$ chords. Voss's conjecture.

$$\lim_{n\to\infty}g_k(n)=\infty.$$

This asks whether criticality forces increasingly chord-rich odd cycles as the order grows. The paper states that its results disprove the conjecture for every fixed $k\geq4$, so it is refuted.