Complete Disjoint coNP-Pairs but no Complete Total Polynomial Search Problems Relative to an Oracle

Consider the following conjectures: H1: the set TFNP of all total polynomial search problems has no complete problems with respect to polynomial reductions. H2: there exists no many-one complete disjoint coNP-pair. We construct an oracle relative to which H1 holds and H2 does not hold. This partially answers a question by Pudl\'ak [Pud17], who lists several hypotheses and asks for oracles that show corresponding relativized hypotheses to be different. As there exists a relativizable proof for the implication H1 -> H2 [Pud17], the relativizations of the hypotheses H1 and H2 are neither independent nor equivalent.