On the existence of strong proof complexity generators

The working conjecture from K'04 that there is a proof complexity generator hard for all proof systems can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows: * There exist a p-time function $g$ stretching each input by one bit such that its range $rng(g)$ intersects all infinite NP sets. We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results) and the range avoidance problem, to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from K'09 is a good candidate for $g$.