In your scenario, you mention a 'debt crisis' but that could be different things:
a) the US going cap-in-hand to its creditors, pleading poverty, and seeking to restructure its debt with promises of immediate fiscal puritanism, involving constitutional amendments that required the running of a sound currency, no matter what, and involving the asset stripping of certain government-backed banks, auto-companies etc;
b)money printing continues, bailouts and stimulus continue, new taxes are raised, unemployment continues to rise, a major western country- the UK is an obvious candidate- goes to the IMF, then the US misses a repayment...
In scenario a) gold would probably drop, after an initial panic spike during negotiations, as there really would be dollar deflation. In b), however, given a bad enough smash, the dollar could become de facto worthless. If this happened, I suppose you could say that the dollar had been inflated out of existence. Gold would certainly be very valuable.
I suppose it boils down to luck, and to politicians making judgements about what will keep them in power and in the good graces of historians.