Linear degree-boundedness conjecture for pivot-minor exclusions

For a positive integer $s$, a graph is $K_{s,s}$-free if it contains no subgraph isomorphic to $K_{s,s}$. Linear pivot-minor degree-boundedness conjecture. For each bipartite graph $H$, there exists a constant $C_H>0$ such that every $K_{s,s}$-free graph excluding $H$ as a pivot-minor has average degree at most $C_Hs$. This is the proposed pivot-minor analogue of Du and McCarty's vertex-minor conjecture; it is stated among the paper's open problems and no resolution is supplied.