# Does high connectivity of line graph of $G$ imply high (cyclic) connectivity of $G$?

All graph considered here are finite, simple and undirected.
We know that a graph $$G$$ is $$k$$-edge connected if and only if its line graph is $$k$$-connected (where $$k\in\mathbb{N}$$). In particular, if $$G$$ is $$k$$-connected, then so is the line graph of $$G$$.

Let $$G$$ be a cubic graph. We know that the line graph of $$G$$ is 4-regular. If the line graph of $$G$$ is 3-connected, then $$G$$ is 3-connected as well. Suppose that the line graph of $$G$$ is 4-connected. Does that mean anything to $$G$$ more than just "$$G$$ is 3-connected"? (perhaps, something similar to high cyclic edge-connectivity?).