Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.
Question What are the conjectures in your field proved or disproved (counterexample found) in recent years, which are noteworthy, but not so famous outside your field?
Answering the question you are welcome to give some comment for outsiders of your field which would help to appreciate the result.
Asking the question I keep in mind by "recent years" something like a dozen years before now, by a "conjecture" something which was known as an open problem for something like at least dozen years before it was proved and I would say the result for which the Fields medal was awarded like a proof of fundamental lemma would not fit "not so famous", but on the other hand these might not be considered as strict criteria, and let us "assume a good will" of the answerer.
Best Answer
A remarkable example is the Gaussian correlation conjecture (which only recently became the Gaussian correlation inequality). The formulation is very simple:
It was formulated over 60 years ago (in the above general form, in 1972) and since then had been attacked by many mathematicians. Despite its apparent simplicity, only several partial results had been obtained before its complete proof in 2014.
What is remarkable is that the proof was quite simple and came from a retired statistician Thomas Royen, whose previous scientific output was not very noticeable. Moreover, the article was turned down by some scientists. It seems that the true reasons were that the author was not well known, and the article itself did not look serious (you can find its first non-LaTeX version here). Finally, it was published by some predatory "Far East" journal. Unsurprisingly, it took about two years for the proof to come to the public attention, and for its author to become famous.
Unfortunately, the story brings out some unpleasant features of the scientific community: hypocrisy and prejudice.
More on the story here.