Conjectured sharp upper bounds for induced degenerate subgraphs of planar graphs

Let $\bar\alpha_d$ denote the minimum, over planar graphs, of the maximum order of a $d$-degenerate induced subgraph divided by the number of vertices. The octahedron and icosahedron give upper bounds for the cases $d=2,3,4$. The sharp-bound conjecture.

$$\bar\alpha_2=2/3,\qquad \bar\alpha_3=5/6,\qquad \bar\alpha_4=11/12.$$

The source proves the lower bound corresponding to $d=3$ only at $3/4$, while the displayed values are witnessed as upper bounds by the cited extremal planar graphs; the conjecture remains open according to the supplied status evidence.