Conjecture on consecutive cycle lengths in 3-connected graphs

Let $k\ge 4$ be an integer, and let $G$ be a $3$-connected non-bipartite graph with minimum degree at least $k$.

Conjecture on consecutive cycle lengths. Except when $G$ is $K_{k+1}$, the graph $G$ contains $k$ cycles of consecutive lengths.

The paper proves this statement for $k\ge 6$ and notes that its proof also handles some special cases for $k=4$ or $k=5$. The conjecture asks whether the condition $k\ge 6$ can be relaxed to $k\ge 4$.