Bounded-gap conjecture for strong-immersion-free clustered coloring

For a graph $H$, let $\chi_*(H)$ denote the strong-immersion coloring parameter used in the paper, and let the clustered chromatic number of a graph class be the least number of colors for which some uniform finite bound exists on the size of every monochromatic component. A graph contains $H$ as a strong immersion when the immersion model satisfies the strongness condition that branch vertices do not occur internally on the routing paths. Strong-immersion clustered-coloring conjecture. There exists a positive integer $C$ such that for every graph $H$, the clustered chromatic number of the class of graphs that do not contain $H$ as a strong immersion is at most $\chi_*(H)+C$. The paper explains that the gap between the clustered chromatic numbers for immersion-free and strong-immersion-free graphs is unknown and conjectures that it is bounded by an absolute constant.