Induced Minors and Region Intersection Graphs

We show that for any positive integers $g$ and $t$, there is a $K_6^{(1)}$-induced-minor-free graph of girth at least $g$ that is not a region intersection graph over the class of $K_t$-minor-free graphs. This answers in a strong form the recently raised question of whether for every graph $H$ there is a graph $H^{\mathrm{\prime }}$ such that $H$-induced-minor-free graphs are region intersection graphs over $H^{\mathrm{\prime }}$-minor-free graphs.