  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 $d = \Omega( \log_{n/k} n)$, (2) when $\mathrm{gap}:=$ the
spectral-gap of a single walk is $ O ( 1/\log^4 n) $ and $k \ge n^{\Omega(1)}$,
(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 $k \ge n^{\Omega (1)}$. 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.
