In the first statement, you want the last $d_0$ to be a $d$?

The answer to the question linked by @AliTaghavi gives an example when $M$ is a manifold with boundary. Namely a symplectic filling of a hyperbolic 3-manifold, $Y$, such that $b_1(Y)>0$ and all orientation-preserving self-homeomorphisms of Y induce the trivial map on $H^2(Y)$. That post doesn't have an answer for closed $M$ though...

Whoops yes, too many negatives - thanks!