Physicist's Euler number conjecture says:

If $G \subset SL(n,\mathbb{C})$ is a finite group, $X=\mathbb{C}^n/G$ is the quotient space and $f:Y \rightarrow X$ a crepant resolution (always exists for $n\leq 3$). Then there exists a basis of $H^*(Y,\mathbb{Q})$ consisting of algebraic cycles in one-to-one correspondence with conjugacy classes of $G$.

I have seen some works (by Reid,...) which date back to 2000. What are the recent results around this conjecture?

See : The McKay correspondence for finite sungroups of SL(3,C), by Miles Reid and Yukari Ito.