Conjecture on collecting all but one twelfth of a planar graph

Let $G$ be a planar graph. To delete a vertex means to remove it and its incident edges. To collect a vertex means to remove it when its current degree is at most $4$; a set is collectable if its vertices can be collected in some sequence. Collecting conjecture. One can delete at most $1/12$ of the vertices of $G$ so that all remaining vertices can be collected. The collected vertices consequently induce a $4$-degenerate subgraph. The preceding results guarantee deletion of less than $1/9$ of the vertices, while the icosahedron shows that one deleted vertex may be necessary; the source presents the stronger $1/12$ bound as believed optimal and gives no resolution.