Conjecture on consecutive odd cycle lengths in 2-connected non-bipartite graphs

Let $k$ be a positive integer. A graph is 2-connected if deleting any one vertex leaves it connected, and it is non-bipartite if it contains an odd cycle. Cycles have consecutive odd lengths when their lengths are odd and consecutive within the sequence of odd integers.

Conjecture on consecutive odd cycle lengths. If $G$ is a 2-connected non-bipartite graph with minimum degree at least $k+1$, then $G$ contains $\lceil k/2\rceil$ cycles with consecutive odd lengths.

The source presents this as a strengthening of an earlier theorem and notes that it would imply a chromatic-number bound; no resolution is supplied.