Ryjáček's conjecture on locally connected graphs

A finite, simple, undirected graph $G$ is locally connected if, for every vertex $u$ of $G$, the subgraph induced by the neighborhood $N_G(u)$ is connected. It is weakly pancyclic if it has a cycle of every order $\ell$ between its girth $g(G)$ and circumference $c(G)$.

Ryjáček's conjecture. Every locally connected graph is weakly pancyclic.

This conjecture proposes that local connectivity forces a graph to contain cycles of all orders between its shortest and longest cycle. The source states it as motivation for the paper; no resolution is supplied here.