Lévêque et al.'s 4-color conjecture for ISK4-free graphs

All graphs under consideration are finite and simple. A graph is ISK4-free if it contains no induced subdivision of $K_4$. Lévêque et al.'s 4-color conjecture. Every ISK4-free graph is $4$-colorable. Lévêque et al. showed that the chromatic number of ISK4-free graphs is bounded by a constant $c\geq 2^{512}$; subsequent work improved the known upper bound to $8$, while the conjectured bound of $4$ remains open.