The minimum-cycle conjecture for non-b$k$d-colourable graphs

Let $G$ be a graph and let $k$ be a positive integer. A cycle has length $0\bmod k$ when its length is divisible by $k$.

Minimum-cycle conjecture. If $\chi(G)>k$, then $G$ contains at least

$$\frac{(k+1)(k-1)!}{2}$$

cycles of length $0\bmod k$.

The complete graph $K_{k+1}$ has precisely this many cycles of length $0\bmod k$, so the conjecture asserts that it is extremal among graphs that are not $k$-colourable. It strengthens the preceding lower bound of $(k-1)!/2$ cycles and is posed as an open question.