[Math] Problems that are largely believed to be true, but are unresolved

Question

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification?

It seems that Goldbach should be true, but this is based on heuristic justification.

I am looking for conjectures that seem to be true, but where the 'why' is something other then a statistical justification, and I want know what exactly that 'why' is.

Edit: Can you please include the reason it is widely believed to be true in your posting? That is the interesting part

Answer 1

The Riemann hypothesis is largely believed to be true, and further conjectures have been made based on its truth (e.g. statements about the distribution of prime numbers) but no one has ever proved it.