I am interested in the consequences of putting faith in conjectures which have not yet been proved beyond all doubt. Has there ever been important conjectures which when disproved have led to the collapse of other mathematical fields, held up by that conjecture. Consequences, can be in maths or in real life.
For example, Euler's Sum of Powers Conjecture was disproved. However I can't find any consequences of this, as nothing was based on it. In addition, would disproving commonly accepted conjectures such as the abc conjecture or goldbach have any consequences.
The only way to collapse a mathematical field is to disprove it's very foundation. Disproving the foundation of a conditional proof, only disproves it, in the sense that if we don't have a proof via other mathematics, we can't prove it. It's very likely to have happened (I don't know the examples).
I'm not aware of any real implications of most open conjectures. Mathematically there might be a few, but I'm not aware of any math coming out of Goldbach, other than parts of plane geometry, that would fail if it were to fail. Most necessary conditions I know, are well know results for conditions that don't just apply to Goldbach. same with most other conjectures. I can relate Beal's conjecture, Goldbach's conjecture and possibly the abc conjecture to discrete logarithms, but that's more about solving speed than about truth.