It's older than 20 years but otherwise the paper where Kashiwara and Vergne introduced their conjecture definitely qualify https://link.springer.com/article/10.1007/BF01579213

There is an important theorem of Duflo stating that the PBW isomorphism for a finite dimensional Lie algebra (in characteristic 0) can be twisted so that its restriction to the invariant part actually become an **algebra** isomorphism $U(\mathfrak g)^{\mathfrak g}\cong S(\mathfrak g)^{\mathfrak g}$. 

The original proof is complicated and require a lot of case by case study. Kashiwara and Vergne suggest a very natural uniform proof of that result. Roughly speaking they postulate that the pull-back of the product on $U(\mathfrak g)$ to $S(\mathfrak g)$ through this twisted version of the PBW isomorphism can be written as $m\circ F$ where $m$ is the multiplication of $S(\mathfrak g)$ and $F$ is an automorphism of a very special form. Then they "rescale" it by introducting a parameter $t$, and observe that Duflo's theorem would follow from the existence of such an $F$ satisfying a certain differential equation w.r.t the parameter $t$. They also observe this would in fact imply a stronger statement, giving an isomorphism between invariant distribution on $\mathfrak g$ and its group $G$ with small suport. Their conjecture is thus really important in harmonic analysis and representation theory.

The main result of their paper is a proof of that conjecture in the solvable case, which can be done in a fairly elementary way. After that, several special cases were proven, but a general proof was given only in 2005 (by Alekseev--Meinrenken) and uses some high level machinery related to Kontsevich's proof of his deformation quantization theorem. Since then this now theorem has been connected to a surprisingly wide range of topics, including Grothenideck-Teichmueller theory, Etingof--Kazhdan theorem, quantum topology and the study of the Atiyah--Bott--Goldman Poisson structure on moduli spaces of flat connections.