Well if they had the ability to prove or disprove some important conjecture, they probably can also prove many smaller results first. Proving a few non controversial results, but ones that are still quite hard to prove would establish their credentials as a mathematician. And if they can't prove something like this, then the chance they have proved some outstanding unsolved conjecture must surely be quite low.

Good advice about it here:

Advice for amateur mathematicians

Some areas of maths do have many contributions from amateur mathematicians even today. Robert Ammann made many contributions to the theory of non periodic tilings. And recreational maths is a fertile field for amateur mathematicians. I.e. the mathematics of games and puzzles.

These links might also be of interest: List of amateur mathematicians

What recent discoveries have amateur mathematicians made?

And - if anyone does try to get the attention of a professional mathematician - do be understanding if they ignore you.

Remember that most mathematicians have the experience of thinking they have proved a result, then finding a flaw in it.

I remember when I was studying for my doctorate in Oxford then for several days I thought I had proved a quite significant result in combinatorics - I mean - not a big one that everyone would have heard of, but an interesting one all the same. I can't remember what it was now - this was perhaps 30 years go. But I remember that I had hit on what seemed a really neat solution. I got as far as telling a few mathematician friends about it and explaining the proof to them - but then to my embarrassment found a rather elementary error on the first page.

Which sadly couldn't be saved.

Sometimes your error leads to either an alternative proof or a proof of something else. But this just blew it apart completely and the following several pages of tight reasoning was useless.

Sometimes even false "proofs" get published. Like the early "proofs" of the four colour theorem in geometry by Kempe (1879) and Tait (1880). Back then the pace of maths was less than today and their proofs were accepted unchallenged for eleven years before someoen found a flaw in them. Four color theorem - history

You can read about Kempe's "proof" of the four colour theorem here: Lab 3 - looks at Kempe's "proof" of 4 colour theorem and explains the flaw in it

So given that, the default assumption if someone gives you a proof of an important result in maths is that they have made a mistake. It is quite easy to miss a mistake like that in your own work - and  just reading a proof is a long job. You can't just pick up a proof and read it in half an hour or so - especially if it has gaps in it and is poorly expressed.

And - it's good to talk about your attempted proofs with other mathematicians, as they often find flaws in them that you can't see yourself.