Sudakov–Verstraëte chromatic-number conjecture for consecutive cycle lengths

For $k\geq 2$, let $\chi_k$ be the largest chromatic number of a graph that does not contain $k$ cycles of consecutive lengths. Sudakov–Verstraëte's conjecture. For every integer $k\geq 2$, $\chi_k=k+1$. The lower bound $\chi_k\geq k+1$ follows from the complete graph $K_{k+1}$; the paper gives no resolution status for the asserted equality in the supplied text.