Malkevitch's consecutive-cycle conjecture for 4-connected planar graphs

Let $G$ be a 4-connected planar graph on $n$ vertices. A cycle of length $4$ is a cycle with four vertices. Malkevitch's conjecture. If $G$ contains a cycle of length $4$, then $G$ contains a cycle of length $\ell$ for every $\ell\in\{n,n-1,\ldots,3\}$. This conjecture concerns cycles of consecutive lengths; the source presents it as posed by Malkevitch in 1988 and does not report a resolution.