I am going to submit the Baum-Connes conjecture because probably nobody else will and I believe its importance is quite understated. The conjecture asserts that the assembly map from the K-homology of the appropriate classifying space of a group $G$ to the K-theory of the reduced group C*-algebra of $G$ is an isomorphism for every group $G$. A proof would immediately imply two old conjectures in completely different areas of mathematics:

• Injectivity implies the Novikov conjecture in high-dimensional topology
• Surjectivity implies the Kadison-Kaplansky conjecture in analysis

Aside from that, the conjecture is deeply linked to a variety of other open problems in differential geometry and topology, for instance in the theory of positive scalar curvature invariants and eta invariants. A bit more speculatively, the conjecture suggests a number of deep dualities in representation theory which could shed light on open problems in that area as well; see, for instance, Aubert-Baum-Plyman's recent work on local Langlands.

Finally, unlike with many conjectures in mathematics, a counter-example would in some ways be more exciting than a proof. Some experts think that $SL(\mathbb{Z},3)$ is a counter-example to surjectivity, but proving it is well beyond the reach of current techniques. Counter-examples to injectivity might require completely new ways to construct discrete groups, and would undoubtedly have applications to geometric group theory.