Solution to a problem of Grünbaum on the edge density of $4$-critical planar graphs

We show that $\mathrm{lim\hspace{0.17em}sup}\mathrm{\mid }E(G)\mathrm{\mid }\mathrm{/}\mathrm{\mid }V(G)\mathrm{\mid }=2.5$ over all $4$-critical planar graphs $G$, answering a question of Grünbaum from 1988.