Constant-factor upper bound for the indicated chromatic number

Determine whether there exists a universal constant C such that every graph G satisfies \(\chi_i(G) \le C\chi(G)\), where \(\chi_i(G)\) is the indicated chromatic number and \(\chi(G)\) is the chromatic number.

Background

The paper studies the indicated chromatic number $\chi_i(G)$, arising from the indicated colouring game, and compares it with the ordinary chromatic number $\chi(G)$. Previously known constructions gave graphs with arbitrarily large chromatic number for which the ratio $\chi_i(G)/\chi(G)$ is at least approximately $3/2$. The authors improve the known lower-ratio examples but do not determine whether the indicated chromatic number is universally bounded by a constant multiple of the chromatic number.