Oliveira conjectured that the order of the mixing time of the exclusion process with $k$-particles on an arbitrary $n$-vertex graph is at most that of the mixing-time of $k$ independent particles. We verify this up to a constant factor for $d$-regular graphs when each edge rings at rate $1/d$ in various cases: (1) when $d = Ω( \log_{n/k} n)$, (2) when $\mathrm{gap}:=$ the spectral-gap of a single walk is $ O ( 1/\log^4 n) $ and $k \ge n^{Ω(1)}$, (3) when $k \asymp n^{a}$ for some constant $0<a<1$. In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of $O(\log n \log \log n / \mathrm{gap})$, which is within a $\log \log n$ factor from Oliveira's conjecture when $k \ge n^{Ω(1)}$. As applications we get new mixing bounds: (a) $O(\log n \log \log n)$ for expanders, (b) order $ d\log (dk) $ for the hypercube $\{0,1\}^d$, (c) order $(\mathrm{Diameter})^2 \log k $ for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.