ET-to-SET conjecture for generic glocal families

Let a glocal family have the extension triviality property (ET), meaning that its glocal extensions are moderately equivalent near the relevant large bifurcation support to trivial extensions. ET-to-SET conjecture. Generic glocal families with the ET property have the strong extension triviality property (SET). The SET property expresses that the auxiliary parameter can be preserved by the base homeomorphism; the conjecture concerns whether genericity upgrades ET to this stronger form.