dg.differential geometry – Homotoping diffeomorphism to a $J$-holomorphic one


Let $M$ be a closed simply-connected smooth manifold. Assume $M$ admits at least one almost complex structure.

Is any diffeomorphism $Mto M$ homotopic as a continuous map to a $J$-holomorphic diffeomorphism $Mto M$ where $J$ is some almost complex structure?