The Connes embedding conjecture, formulated in 1976, asserts that every type II₁ von Neumann factor embeds into an ultrapower of the hyperfinite II₁ factor. Even though it is sometimes known as the Connes embedding “problem”, my understanding is that it was widely believed to be true. This was shown in 2011–2013 to be equivalent to a form of Cirel'son's problem about the equality of two convex sets defined by quantum Bell inequalities (often denoted $C_{\mathrm{qa}}(n,k) \subseteq C_{\mathrm{qc}}(n,k)$ for $n,k$ integers; see here¶2.3–2.7 for a possible definition, but very roughly $C_{\mathrm{qc}}$ is the set of quantum correlations defined by commuting projections in an arbitrary Hilbert space, whereas $C_{\mathrm{qa}}$ is the closure of $C_{\mathrm{q}}$ defined analogously for finite-dimensional Hilbert spaces).
In 2020, Ji, Natarajan, Vidick, Wright and Yuen put forth a preprint called $\mathrm{MIP}^* = \mathrm{RE}$ proving, for computability-theory related reasons, that in fact $C_{\mathrm{qa}} \neq C_{\mathrm{qc}}$, and thus refuting Connes's embedding conjecture.
Quanta magazine wrote a piece about it.
(In a further twist, here recounted in one of the authors' blog, an error was then found in one of the papers on which the Ji & al. paper relies. My understanding is that this does not affect the main result: see here for the fix; however perhaps as a result of this the preprint does not appear to be published as such but in an ongoing series of papers. Will someone please edit this answer if this is not an accurate assessment of the situation.)